Origin of Magic Angular Momentum in a Quantum Dot under Strong Magnetic Field

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介绍陀螺的英语作文

介绍陀螺的英语作文英文回答：Introduction to Gyroscope.Gyroscopes are fascinating devices that have been used in various applications for centuries. A gyroscope is essentially a spinning wheel or disk that maintains its orientation regardless of any external forces acting on it. This unique property makes gyroscopes incredibly useful in a wide range of fields, from navigation systems to toys.One of the most common applications of gyroscopes is in stabilizing devices such as drones and cameras. By using gyroscopes, these devices can maintain a steady position even when they are subject to external disturbances. For example, when flying a drone in windy conditions, the gyroscope helps to keep the drone level and prevent it from tipping over.Another interesting application of gyroscopes is in the field of physics education. Gyroscopes are often used to demonstrate principles of angular momentum and precessionin a hands-on way. By playing with a gyroscope, students can see firsthand how the spinning motion of the device affects its stability and orientation.In the world of sports, gyroscopes are also used to improve athletic performance. For example, some athletes use gyroscopic exercise tools to strengthen their wrists and improve their coordination. By incorporating gyroscopes into their training regimen, athletes can enhance their balance and stability, which can translate to better performance on the field or court.Overall, gyroscopes are versatile devices that play a crucial role in many different industries. Whether it's in stabilizing equipment, educating students, or enhancing athletic performance, gyroscopes continue to be a valuable tool in our modern world.中文回答：陀螺是一种非常有趣的装置，几个世纪以来一直被用于各种应用领域。

symmetric 名词

Symmetric1. IntroductionSymmetric is a term used in mathematics and other fields to describe objects or concepts that possess a certain type of symmetry. In mathematics, symmetry refers to a property where an object remains unchanged under certain transformations, such as reflection, rotation, or translation. In this article, we will explore the concept of symmetry and its various applications in different disciplines.2. Symmetry in MathematicsSymmetry plays a fundamental role in mathematics and is widely studied in various branches, including geometry, algebra, and group theory. In geometry, symmetric objects are those that can be divided into two or more parts that are mirror images of each other. For example, a circle is symmetric with respect to any line passing through its center.Symmetry can also be observed in patterns and shapes. Regular polygons such as squares and equilateral triangles possess rotational symmetry because they can be rotated by certain angles without changing their appearance. Fractals, which are intricate mathematical patterns that repeat at different scales, often exhibit self-similarity and possess various types of symmetries.In algebra, symmetry is explored through the concept of functions. A function is said to be symmetric if it satisfies the condition f(x) =f(-x) for all values of x in its domain. This property implies that the graph of the function is symmetric with respect to the y-axis.Group theory provides a rigorous framework for studying symmetry by defining mathematical structures called groups. A group consists of a set of elements and an operation that combines two elements to produce another element in the set. Symmetry groups describe all possible symmetries of an object or system and play a vital role in understanding its properties.3. Symmetry in PhysicsSymmetry has profound implications in physics and is essential for understanding the laws of nature. The principle of symmetry lies at theheart of many fundamental theories, such as classical mechanics, quantum mechanics, and general relativity.In classical mechanics, the conservation of angular momentum is a consequence of rotational symmetry. The laws of motion remain the same regardless of the direction in which an object is oriented. This symmetry is evident in everyday life, where the behavior of objects under different orientations follows consistent patterns.In quantum mechanics, symmetry plays a crucial role in determining the behavior of particles and their interactions. The principles of quantum field theory rely heavily on the concept of gauge symmetry, which describes the invariance of physical laws under certain transformations. Symmetries such as time translation symmetry and particle-antiparticle symmetry are fundamental to our understanding of elementary particles and their interactions.General relativity, Einstein’s theory of gravity, incorporates the principle of general covariance, which states that physical laws should be independent of the choice of coordinates. This symmetry reflects the idea that space and time are not absolute but rather depend on the observer’s frame of reference.4. Symmetry in BiologySymmetry is prevalent in biology and plays a crucial role in understanding various biological processes and structures. Many organisms exhibit bilateral symmetry, where their bodies can be divided into two mirror-image halves along a plane. This type of symmetry is observed in animals ranging from insects to mammals.Bilateral symmetry provides advantages such as improved mobility and sensory perception. It allows for efficient movement by dividing body parts into corresponding pairs such as legs or wings. It alsofacilitates better coordination between sensory organs located on opposite sides of an organism.Symmetry is also observed at smaller scales within organisms. For example, DNA molecules possess helical symmetry due to their double-stranded structure. Proteins often exhibit internal symmetries that influence their folding patterns and functions.Studying symmetric patterns in nature can provide insights into evolutionary processes and ecological relationships between differentspecies. Understanding how symmetrical structures arise and function can help biologists unravel complex biological systems.5. Symmetry in Art and DesignSymmetry has been a fundamental principle in art and design for centuries. Artists and designers often use symmetrical patterns and compositions to create aesthetically pleasing and visually balanced works.Symmetry can be found in various art forms, including painting, sculpture, architecture, and textiles. The use of bilateral symmetry can create a sense of harmony and orderliness. Radial symmetry, where elements are arranged around a central point, is also frequently employed to achieve balance and visual interest.In modern design, the concept of symmetry has evolved to include asymmetrical compositions as well. Asymmetry introduces a dynamic element by intentionally breaking traditional symmetrical arrangements. This approach adds visual tension and can create a more engaging and thought-provoking experience for the viewer.6. ConclusionSymmetry is a fascinating concept that permeates many aspects of our world, from mathematics to physics, biology, and art. It represents an inherent orderliness and balance that is both aesthetically pleasing and intellectually stimulating.The study of symmetry has led to significant advancements in various fields, providing insights into the fundamental laws of nature, the structure of biological systems, and the principles of design. By understan ding symmetry’s underlying principles and exploring its applications, we gain a deeper appreciation for the intricate patterns that surround us.。

05-06(2)《大学物理(A)I》期末考试试卷(双语)(滕小瑛)

2. Mass is inertia for linear motion. What is rotational inertia? A) mass B) torque C) I / M D) Moment of inertia []
3. Two sinusoidal waves travel in the same direction and have the same frequency. Their amplitudes are y1m and y2m. The smallest possible amplitude of the resultant wave is: A) y1m + y2m and occurs when they are 180° out of phase B) ⎜y1m – y2m⎜ and occurs when they are 180° out of phase C) y1m + y2m and occurs when they are in phase D) ⎜y1m – y2m⎜ and occurs when they are in phase []
8. A standing wave pattern is established in a string as shown. The wavelength of one of the component traveling waves is: _______ _________.
9.. In theYoung’s double-slit system, if one of slits is covered up by a media layer （媒质膜） (n=1.40) and the other is covered up by a media layer (n=1.70), (they have same thickness). After the two media layers are inserted, the original central bright fringe will be replaced by the 7-order bright fringes (屏幕上原中央极大所在点变为第七级明纹), if the wavelength of incident light is 450nm, The thickness of both two layers is__________________________。

angular momentum公式

angular momentum公式The vector product and torqueThe torque involves the two vectors and,and its direction is perpendicular to the plane of and.We can establish a mathematical relationship between,and,using a new mathematical operation called the vector product,or cross product:The formal definition of the vector product:the magnitude is:where is the angle between and.的方向遵循右手定则。

Some properties of the vector product:If is parallel to(),then;If is perpendicular to,thenThe cross product of any two vectors and can be expressed in the following determinant form:Angular momentum of a particleThe instantaneous angular momentum:The magnitude of iswhere is the angle between and.It follows that is zero when is parallel to().In other words,when the linear velocity of the particle is along a line that passes through the origin,theparticle has zero angular momentum with respect to the origin.On the other hand,if is perpendicular to(),then.At that instant,the particle moves exactly as if it were on the rim of a wheel rotating about the origin.假设滑冰的人划过一个杆子，如果用手抓杆子，就会出现对应两种情况，常见的情形就是人绕着杆子打转。

转动惯量坐标系转换公式

转动惯量坐标系转换公式English Answer:Rotational Inertia Coordinate Transformation Formula.The rotational inertia tensor is a physical quantity that characterizes the resistance of an object to angular acceleration. It is a tensor of rank 2, and its components are defined as the sum of the products of the masses of the particles in the object and the squares of their distances from the axis of rotation.The rotational inertia tensor is a useful quantity for describing the motion of rigid bodies. For example, it can be used to calculate the angular momentum of a body, its kinetic energy, and its moment of inertia.The rotational inertia tensor is also a tensor, which means that it transforms in a specific way under coordinate transformations. The transformation formula for therotational inertia tensor is given by:I' = R I R^T.where:I is the rotational inertia tensor in the original coordinate system.I' is the rotational inertia tensor in the new coordinate system.R is the rotation matrix that transforms the original coordinate system into the new coordinate system.The rotation matrix is a 3x3 matrix that describes the rotation of the original coordinate system into the new coordinate system. The elements of the rotation matrix are the cosines of the angles between the axes of the two coordinate systems.The rotational inertia tensor is a symmetric matrix,which means that it is equal to its transpose. This implies that the transformation formula for the rotational inertia tensor can be simplified to:I' = R I R.The rotational inertia tensor is a useful quantity for describing the motion of rigid bodies. It can be used to calculate the angular momentum of a body, its kinetic energy, and its moment of inertia. The transformation formula for the rotational inertia tensor is given by:I' = R I R^T.where R is the rotation matrix that transforms the original coordinate system into the new coordinate system.Chinese Answer:转动惯量坐标系转换公式。

taylor proudman定理

taylor proudman定理The Taylor Proudman theorem, also known as the Taylor Proudman effect, is a principle in fluid dynamics that describes the behavior of flow in a rotating system. It states that in a steady-state, slow-flowing, and highly turbulent rotating fluid, there will be a set of parallel rotating flow patterns, known as Taylor columns or Taylor vortices, that do not mix with each other.This theorem is a consequence of the conservation of angular momentum in a rotating system. In a rotating fluid, the Coriolis force causes fluid particles to experience a perpendicular acceleration, resulting in the formation of columns or vortices aligned with the axis of rotation.The Taylor Proudman theorem has significant applications in various fields of study, including atmospheric and oceanic dynamics, astrophysics, and engineering. For example, in atmospheric science, it helps explain the formation and behavior of large-scale atmospheric circulation patterns such as jet streams. In oceanography, it plays a crucial role in understanding the dynamics of ocean currents and the formation of eddies.Overall, the Taylor Proudman theorem provides essential insights into the dynamics of rotating fluid systems, allowing scientists and engineers to better understand and predict fluid behavior in various scenarios.。

角动量守恒定律原理

角动量守恒定律原理The principle of conservation of angular momentum is a fundamental concept in physics that states that the total angular momentum of a system remains constant if no external torque acts on it. 角动量守恒定律是物理学中的一个基本概念，如果没有外部力矩作用于系统，那么系统的总角动量保持不变。

This principle can be applied to a wide range of physical phenomena, from the motion of celestial bodies to the spinning of a figure skater. 这个原理可以应用于广泛的物理现象，从天体的运动到花样滑冰运动员的旋转。

For example, when a figure skater pulls their arms in during a spin, their moment of inertia decreases, causing their angular velocity to increase in order to conserve angular momentum. 例如，当一个花样滑冰运动员在旋转时，缩小臂部尺寸，使其惯性矩减小，角速度增加以保持角动量。

Similarly, the conservation of angular momentum can explain why ice skaters are able to perform complex spins and rotations with suchgrace and precision. 同样地，角动量守恒定律可以解释为什么冰上运动员能如此优雅、精准地进行复杂的旋转。

介绍陀螺的英文作文

介绍陀螺的英文作文英文：Gyroscope is a device that is used to measure or maintain orientation and angular velocity. It is commonly used in navigation systems, aircrafts, and even in toys. The gyroscope works on the principle of conservation of angular momentum, which means that a spinning object will maintain its orientation and angular velocity unless acted upon by an external force.There are different types of gyroscopes, such as the mechanical gyroscope, the fiber optic gyroscope, and the ring laser gyroscope. The mechanical gyroscope is the oldest type and is still used in some applications. It consists of a spinning wheel or disk that is mounted on a gimbal, which allows it to rotate freely in any direction. The fiber optic gyroscope and the ring laser gyroscope are more modern types, which use light to measure rotation.One of the most common uses of gyroscopes is in stabilizing cameras and drones. By using a gyroscope, the camera or drone can maintain a steady orientation and avoid shaky footage. Another example is in smartphones, where gyroscopes are used to detect the orientation of the device and adjust the screen accordingly.Overall, the gyroscope is an important device in many applications and has greatly improved the accuracy and stability of various systems.中文：陀螺是一种用于测量或维持方向和角速度的装置。

来自英文网站的专业气象词汇

关于天气，天气预报的一些词汇和解释（来自英文网站/er/box/glossary.htm）Terms used by meteorologists, forecasters, weather observers,and in weather forecastsAblation- The process of being removed. Snow ablation usually refers to removal by meltingAbsolute Humidity- The density of water vapor. It is the mass of the water vapor divided by the volume that it occupies.Accretion- Growth of precipitation particles by collision of ice crystals with super cooled liquid droplets which freeze on impactAccessory Clouds- Clouds that are dependent on a larger cloud system for development and continuance. Accessory clouds associated with the thunderstorm include roll, shelf, mammatus, and wall clouds.Acid Rain- Cloud or rain droplets containing pollutants, such as oxides of sulfur and nitrogen, to make them acidic (e.g. pH < 5.6).Additive data- A group of coded remarks in a weather observation that includes pressure tendency, amount of precipitation, and maximum/minimum temperature during specified periods of time.Adiabatic- changes in temperature caused by the expansion (cooling) or compression (warming) of a body of air as it rises or descends in the atmosphere.Adiabatic Process- The change of temperature of air without transferring heat. In an adiabatic process compression results in warming, and expansion results in cooling.Advection- The horizontal transport of air, moisture or other atmospheric properties. Commonly used with temperatures, i.e., "warm air advection".Advection Fog- a type of fog that results from the advection of moist air over a cold surface and the cooling of the air to its dew point that follows; this type of fog is most common in coastal regions.Advisory- Advisories are issued for weather situations that cause significant inconveniences but do not meet warning criteria and, if caution is not exercised,could lead to life-threatening situations. Advisories are issued for significant events that are occurring, are imminent, or have a very high probability of occurrence.Aerosol- Particles of matter, solid or liquid, larger than a molecule but small enough to remain suspended in the atmosphere (up to 100? m diameter). Natural origins include salt particles from sea spray and clay particles as a result of weathering of rocks. Aerosols can also originate as a result of man's activities and in this case are often considered pollutants.Aerovane- Aerovanes are commonly used at many weather stations and airports to measure both wind direction and speed. They are similar to wind vanes and cup anemometers except have three-bladed propellers attached to the end of the vane.AFOS- the Automation of Field Operations and Services; AFOS is the computer system that links National Weather Service offices and other computer networks, such as the NOAA Weather Wire, to transmit weather information.AGL- above ground level.Air- the mixture of gases that make-up the earth's atmosphere.Air Mass- A large body of air that has similar horizontal temperature and moisture characteristics.Air-mass Thunderstorm- Generally, a thunderstorm not associated with a front or other type of synoptic-scale forcing mechanism. Air mass thunderstorms typically are associated with warm, humid air in the summer months; they develop during the afternoon in response to insolation, and dissipate rather quickly after sunset.Air Parcel- An imaginary small body of air that is used to explain the behavior of air. A parcel is large enough to contain a very great number of molecules, but small enough so that the properties assigned to it are approximately uniform throughout.Air Pollution- The existence in the air of substances in concentrations that are determined unacceptable. Contaminants in the air we breathe come mainly from manufacturing industries, electric power plants, automobiles, buses, and trucks.Air Pressure- (atmospheric pressure) air pressure is the force exerted on asurface by the weight of the air above it. The internationally recognized unit for measuring this pressure is the kilopascal.Airstream- A significant body of air flowing in the same general circulation. Albedo- The percentage of light reflected by an object. Snow covered areas have a high albedo (0.9 or 90%) due to their white color.Alberta Clipper- A small, fast-moving low-pressure system that forms in western Canada and travels southeastward into the United States. These storms, which generally bring little precipitation, generally precede an Arctic air mass.Altimeter- An active instrument (see active system) used to measure the altitude of an object above a fixed level.Altimeter setting- That pressure value to which an aircraft altimeter scale is set so that it will indicate the altitude above mean sea-level of an aircraft on the ground at the location for which the value was determined.Altitude- Height expressed as the distance above a reference point, which is normally sea level or ground level.Altocumulus- Mid-altitude clouds with a cumuliform shape.Altostratus- Mid-altitude clouds with a flat sheet-like shape.Anabatic- wind flowing up an incline, such as up a hillside; upslope wind. Anafront- A front at which the warm is ascending the frontal surface up to high altitudes.Anemometer- An instrument that measures wind speed.Aneroid barometer- An instrument built around a metal structure that bends with changing air pressure. These changes are recorded on a pointer that moves back and forth across a printed scale.Angular Momentum- the energy of motion of a spinning body or mass of air or water.Angular Velocity- the rate at which a spinning body rotates.Anomaly- The deviation of (usually) temperature or precipitation in a given region over a specified period from the normal value for the same region.Anticyclone- A large body of air in which the atmospheric pressure is higher than the pressure in the surrounding air. The winds blow clockwise around an anticyclone in in the Northern Hemisphere.Anticyclonic- describes the movement of air around a high pressure, and rotation about the local vertical opposite the earth's rotation. This is clockwise in the Northern Hemisphere.Anvil Cloud- The flat, spreading top of a Cb (cumulonimbus), often shaped like an anvil. Thunderstorm anvils may spread hundreds of miles downwind from the thunderstorm itself, and sometimes may spread upwind (seeback-sheared anvil).Anvil Crawler - A lightning discharge occurring within the anvil of a thunderstorm, characterized by one or more channels that appear to crawl along the underside of the anvil. They typically appear during the weakening or dissipating stage of the parent thunderstorm, or during an active MCS.Anvil Dome - A large overshooting top or penetrating top.Anvil Rollover - A circular or semicircular lip of clouds along the underside of the upwind part of a back-sheared anvil, indicating rapid expansion of the anvil. See cumuliform anvil, knuckles, mushroom.Anvil Zits - Frequent (often continuous or nearly continuous), localized lightning discharges occurring from within a thunderstorm anvil.Arctic Air- a mass of very cold, dry air that usually originates over the Arctic Ocean north of Canada and Alaska.Arctic High- a very cold high pressure that originates over the Arctic Ocean.Arcus- A low, horizontal cloud formation associated with the leading edge of thunderstorm outflow (i.e., the gust front). Roll clouds and shelf clouds both are types of arcus clouds.Aridity- A general term used to describe areas suffering from lack of rain or drought. More specifically, a condition in which evaporation exceeds precipitation.ASOS- Automated Surface Observing System. This system observes sky conditions, temperature and dewpoint, wind direction and speed, and barometric pressure, and precipitation.Atmosphere- The mass of air surrounding the earth and bound to it more or less permanently by the earth's gravitational attraction.Atmospheric Pressure- (also called air pressure or barometric pressure) The pressure asserted by the mass of the column of air directly above any specific point.Atmospheric Stability- An indication of how easily a parcel of air is lifted. If the air is very stable it is difficult to make the parcel rise. If the air is very unstable the parcel may rise on its own once started.Aurora Borealis- Also known as the northern lights - The luminous, radiant emission from the upper atmosphere over middle and high latitudes, and centered around the earth's magnetic poles. These silent fireworks are often seen on clear winter nights in a variety of shapes and colors.Automated Weather Station- An unmanned station with various sensors that measure weather elements such as temperature/wind/pressure and transmit these readings for use by meteorologists.VHRR- Advanced Very High Resolution Radiometer. Main sensor on U.S. polar orbiting satellites.Avalanche- a large mass of rapidly moving snow down a steep mountain slope.AVN- Aviation Model generated every 12 hours by NCEP.AWIPS- Advanced Weather Information Processing System. New NWS computer system integrating graphics, satellite and radar imagery. The successor to AFOS.Back Door Cold Front- A front that moves east to west in direction rather than the normal west to east movement. For instance, one that enters Southern New England from the Gulf of Maine.Back-building Thunderstorm- A thunderstorm in which new development takes place on the upwind side (usually the west or southwest side), such that the storm seems to remain stationary or propagate in a backward direction. Back-sheared Anvil- A thunderstorm anvil which spreads upwind, against the flow aloft. A back-sheared anvil often implies a very strong updraft and a high severe weather potential.Backing Wind- Wind which shifts in a counterclockwise direction with time at a given location (e.g. from southerly to southeasterly), or change direction in a counterclockwise sense with height (e.g. westerly at the surface but becoming more southerly aloft). Backing winds with height are indicative of cold air advection (CAA). The opposite of veering winds.Ball lightning- A relatively rarely seen form of lightning, generally consisting of an orange or reddish ball of the order of a few cm to 30 cm in diameter and of moderate luminosity, which may move up to 1 m/s horizontally with a lifetime of a second or two.Barber Pole- A thunderstorm updraft with a visual appearance including cloud striations that are curved in a manner similar to the stripes of a barber pole. The structure typically is most pronounced on the leading edge of the updraft, while drier air from the rear flank downdraft often erodes the clouds on the trailing side of the updraft.Baroclinic Zone- A region in which a temperature gradient exists on a constant pressure surface. Baroclinic zones are favored areas for strengthening and weakening systems.Barogram- The graphic record of pressure produced by a barograph. Barograph- An instrument that provides a continuous record of atmospheric pressure.Barometer- An instrument for measuring atmospheric pressure.Barometric pressure- The actual pressure value indicated by a pressure sensor.Barometric Tendency- The amount and direction of change in barometer readings over a three-hour period.Barotropic System- A weather system in which temperature and pressure surfaces are coincident, i.e., temperature is uniform (no temperature gradient) on a constant pressure surface. Barotropic systems are characterized by a lack of wind shear, and thus are generally unfavorable areas for severe thunderstorm development.Bear's Cage- A region of storm-scale rotation, in a thunderstorm, which is wrapped in heavy precipitation. This area often coincides with a radar hook echo and/or mesocyclone, especially one associated with an HP storm. The term reflects the danger involved in observing such an area visually, whichmust be done at close range in low visibility.Beaufort Scale- A scale that indicates the wind speed using the effect wind has on certain familiar objects.Beaver('s) Tail- A particular type of inflow band with a relatively broad, flat appearance suggestive of a beaver's tail. It is attached to a supercell's general updraft and is oriented roughly parallel to the pseudo-warm front, i.e., usually east to west or southeast to northwest.Black Ice- thin, new ice that forms on fresh water or dew covered surfaces; it is common on roadways during the fall and early winter and appears "black" because of its transparency.Blizzard- Includes winter storm conditions of sustained winds or frequent gusts of 35 mph or more that cause major blowing and drifting of snow, reducing visibility to less than one-quarter mile for 3 or more hours. Extremely cold temperatures often are associated with dangerous blizzard conditions. Blizzard warning- Issued when blizzard condition are expected or are occurring.Blocking High- A high pressure area (anticyclone), often aloft, that remains nearly stationary or moves slowly compared to west-to-east motion. It blocks the movement eastward movement of low pressure areas (cyclones) at its latitude..Blowing Dust- dust that is raised by the wind to moderate heights above the ground to a degree that horizontal visibility decreases to less than seven miles. Visibilities of 1/8 mile or less over a widespread area are criteria for a Blowing Dust Advisory.Blowing Sand- Sand particles picked up from the surface of the earth by the wind to moderate heights above the ground, reducing the reported horizontal visibility to less than 7 statute miles.Blowing Snow- Wind driven snow that reduces visibility to six miles or less causing significant drifting. Blowing snow may be snow that is falling and/or loose snow on the ground picked up by the wind.Blowing spray- Water droplets torn by the wind from a body of water, generally from the crests of waves, and carried up into the air in such quantities that they reduce the reported horizontal visibility to less than 7 statute miles.Blustery- Descriptive term for gusty winds that accompany cold weather. Bomb Cyclone- An extratropical area of low pressure in which the central pressure drops at least 24 millibars in 24 hours.Boundary Layer- In general, a layer of air adjacent to a bounding surface. Specifically, the term most often refers to the planetary boundary layer, which is the layer within which the effects of friction are significant. For the earth, this layer is considered to be roughly the lowest one or two kilometers of the atmosphere.Bow echo- A radar echo which is linear but bent outward in a bowshape. Damaging straight-line winds often occur near the "crest" or center of a bowecho. Areas of circulation also can develop at either end of a bow echo, which sometimes can lead to tornado formation - especially in the left (usually northern) end, where the circulation exhibits cyclonic rotation.Box (or Watch Box) - A severe thunderstorm or tornado watch.Breezy- Wind in the range of 15 mph to 25 mph with mild or warm temperatures.Brisk- Wind in the range of 15 to 25 mph when the temperature is cold. Broken Clouds- Clouds which cover between 5/8ths and 7/8ths of the sky. Buoyancy- That property of an object that enables it to float on the surface of a liquid, or as in the case with air parcels, to ascend and remain freely suspended in the atmosphere.Bubble High- A mesoscale area of high pressure, typically associated with cooler air from the rainy downdraft area of a thunderstorm or a complex of thunderstorms. A gust front or outflow boundary separates a bubble high from the surrounding air.Bulk Richardson Number (or BRN)- A non-dimensional number relating vertical stability and vertical shear (generally, stability divided by shear). High values indicate unstable and/or weakly-sheared environments; low values indicate weak instability and/or strong vertical shear. Generally, values in the range of around 50 to 100 suggest environmental conditions favorable for supercell development.Bust- An inaccurate forecast, usually a situation in which significant weather is expected, but does not occur.BWER - Bounded Weak Echo Region. (Also known as a vault.) Radar signature within a thunderstorm characterized by a local minimum in radar reflectivity at low levels which extends upward into, and is surrounded by, higher reflectivities aloft. This feature is associated with a strong updraft and is almost always found in the inflow region of a thunderstorm. It cannot be seen visually.CAA- Cold Air AdvectionCalm- the absence of apparent motion in the air.Cap (or Capping Inversion)- A layer of relatively warm air aloft (usually several thousand feet above the ground) which suppresses or delays the development of thunderstorms. Air parcels rising into this layer become cooler than the surrounding air, which inhibits their ability to rise further. As such, the cap often prevents or delays thunderstorm development even in the presence of extreme instability.CAPE- Convective Available Potential Energy. A measure of the amount of energy available for convection. CAPE is directly related to the maximum potential vertical speed within an updraft; thus, higher values indicate greater potential for severe weather. Observed values in thunderstorm environments often may exceed 1,000 joules per kilogram (j/kg), and in extreme cases may exceed 5,000 j/kg. However, as with other indices or indicators, there are no threshold values above which severe weather becomes imminent.Cb- Cumulonimbus cloudCeiling- The height of the lowest layer of broken or overcast clouds. Ceilometer- A device used to evaluate the height of clouds or the vertical visibility into a surface-based obscuration.Cell- Convection in the form of a single updraft, downdraft, orupdraft/downdraft couplet, typically seen as a vertical dome or tower as in a cumulus or towering cumulus cloud. A typical thunderstorm consists of several cellsCelsius- a temperature scale in which zero is the freezing point of water and one hundred is the boiling point.Chance- A 30, 40 or 50 percent chance of occurrence of measurable precipitation.Chinook Wind- A strong downslope wind that causes the air to warm rapidly as a result of compressive heating; called a foehn wind in Europe.Circulation- The pattern of the movement of air. General circulation is the flow of air of large, semi-permanent weather systems, while secondary circulation is the flow of air of more temporary weather systems.Cirriform- High altitude ice clouds with a very thin wispy appearance.Cirrocumulus- Cirrus clouds with vertical development.Cirrostratus- Cirrus clouds with a flat sheetlike appearance.Cirrus- High clouds, usually above 18,000 feet, composed of ice crystals and appearing in the form of white, delicate filaments or white or mostly white patches or narrow bands.Clear- Sky condition of less than 1/10 cloud coverage.Clear Slot- A local region of clearing skies or reduced cloud cover, indicating an intrusion of drier air; often seen as a bright area with higher cloud bases on the west or southwest side of a wall cloud.Climate-The prevalent long term weather conditions in a particular area. Climatic elements include precipitation, temperature, humidity, sunshine and windvelocity and phenomena such as fog, frost, and hail storms. Climate cannot be considered a satisfactory indicator of actual conditions since it is based upon a vast number of elements taken as an average.Climate change- This strictly refers to all forms of climatic inconsistency. But it is often used in a more restricted sense to imply a significant change. Within the media, climate change has been used synonymously with global warming. Scientists, however, use the term in a wider sense to include past climate changes also.Climate Normals- Averages of temperatures, precipitation, snowfall, etc. made over standard 30 year periods. These normals span across 3 decades and are rederived every 10 years.Climatology- the scientific study of climate.Closed Low- A low pressure area with a distinct center of cyclonic circulation which can be completely encircled by one or more isobars or height contour lines. The term usually is used to distinguish a low pressure area aloft from a low-pressure trough. Closed lows aloft typically are partially or completely detached from the main westerly current, and thus move relatively slowly.Cloud- A visible cluster of tiny water and/or ice particles in the atmosphere.Cloud Base- For a given cloud or cloud layer, it is the lowest level in the atmosphere where cloud particles are visible.Cloud Condensation Nuclei- small particles in the air on which water vapor condenses and forms cloud droplets.Cloud Streets- Rows of cumulus or cumulus-type clouds aligned parallel to the low-level flow. Cloud streets sometimes can be seen from the ground, but are seen best on satellite photographs.Cloud Tags - Ragged, detached cloud fragments; fractus or scud.Cloudburst- A sudden, intense rainfall that is normally of short duration.Cloudy- the state of the sky when 7/10ths or more of the sky is covered by clouds.Coastal Flood Warning- Issued when there is widespread coastal flooding expected within 12 hours, more than just typical overwash.Coastal Flooding- The inundation of land areas along the coast caused by sea water above normal tidal actions. This is often caused by prolonged strong onshore flow of wind and/or high astronomical tides.Coastal Forecast- A forecast of wind, wave and weather conditions between the coastline and 25 miles offshore.Coastal Waters- include the area from a line approximating the mean high water along the mainland or island as far out as 25 miles including the bays, harbors and sounds.Cold Advection- (CAA) Transport of cold air into a region by horizontal winds.Cold Air Damming- Cold air damming occurs when a cold dome of high pressure settles over northeastern New England. The clockwise circulation around the high pressure center brings northeasterly winds to the mid Atlanticregion. The northeasterly winds bank cold air against the eastern slopes of the Appalachian Mountains. Warmer air from the west or southwest is lifted above the cold air as it moves instead of warming the surface.Cold-air Funnel- A funnel cloud or (rarely) a small, relatively weak tornado that can develop from a small shower or thunderstorm when the air aloft is unusually cold (hence the name). They are much less violent than other types of tornadoes.Cold Front- A narrow transition zone separating advancing colder air from retreating warmer air. The air behind a cold front is cooler and typically drier than the air it is replacing.Cold Low- a low pressure system with cold air mass from near the surface to all vertical levels (also called a cold core low).Cold Pool- A region of relatively cold air, represented on a weather map analysis as a relative minimum in temperature surrounded by closed isotherms. Cold pools aloft represent regions of relatively low stability, whilesurface-based cold pools are regions of relatively stable air.Collar Cloud- Frequently used as a synonym for wall cloud, although it actually is a generally circular ring of cloud surrounding the upper portion of a wall cloud.Comma Cloud- A synoptic scale cloud pattern with a characteristiccomma-like shape, often seen on satellite photographs associated with large and intense low-pressure systems.Condensation- The process by which water vapor becomes a liquid; the opposite of evaporation, which is the conversion of liquid to vapor.Condensation Nuclei- Small particles in the air around which water vapor condenses.Conduction- The transfer of heat by molecular action between bodies that are in contact.Confluence- A pattern of wind flow in which air flows inward toward an axis oriented parallel to the general direction of flow. It is the opposite of difluence. Confluence is not the same as convergence. Winds often accelerate as they enter a confluent zone, resulting in speed divergence which offsets the (apparent) converging effect of the confluent flow.Congestus (or Cumulus Congestus)- A large cumulus cloud with great vertical development, usually with a cauliflower-like appearance, but lacking the characteristic anvil shaped top of a Cb.Continental Air Mass- A dry air mass originating over a large land area.Contrail- A cloud-like stream formed in cold, clear air behind the engines of an airplane.Convection- The transfer of heat within a the air by its movement. The term is used specifically to describe vertical transport of heat and moisture, especially by updrafts and downdrafts in an unstable atmosphere.Convective Outlook- A forecast containing the area(s) of expected thunderstorm occurrence and expected severity over the contiguous United States, issued several times daily by the SPC.Convective Temperature- The approximate temperature that the air near the ground must warm to in order for surface-based convection to develop, based on analysis of a sounding.Convergence- An atmospheric condition that exists when the winds cause a horizontal net inflow of air into a specified region. Divergence is the opposite, where winds cause a horizontal net outflow of air from a specified region. Coordinated Universal Time (UTC)- The time in the zero degree meridian time zone.Combined Seas- The combined height of swell and wind waves.Cooling Degree Day- A form of degree day used to estimate the required energy for cooling. one cooling degree day occurs for each degree the daily mean temperature is above 65 degrees Fahrenheit.Coriolis Force- An apparent force caused by the rotation of the Earth. In the Northern Hemisphere winds are deflected to the right, and in the Southern Hemisphere to the left. In synoptic scale weather systems (hurricanes and large mid-latitude storms), the Coriolis force causes the air to rotate around a low pressure center in a cyclonic direction. The air flowing around a hurricane spins counter-clockwise in the northern hemisphereCorona- a disk of light surrounding the sun or moon; this is a result of the diffraction of light by small water droplets.CRS- Console Replacement System. This consists of a computer system and computer voice that is used to automate NOAA Weather Radio. Cumulonimbus Cloud- A vertically developed cloud, often capped by an anvil shaped cloud. Also called a thunderstorm cloud, it is frequently accompanied by heavy showers, lightning, thunder, and sometimes hail or gusty winds.Cumulus Cloud- A cloud in the shape of individual detached domes, with a flat base and a bulging upper portion resembling cauliflower.Cumulus Congestus- A large cumulus cloud with great vertical development, usually with a cauliflower-like appearance, but lacking the characteristic anvil shaped top of a Cb.Cut Off Low- An upper level low pressure system that is no longer in the normal west to east upper air flow. Usually a cut-off low will lie to the South of the established upper air flow.Cyclogenesis- Development or intensification of a low-pressure center Cyclone- An area of low pressure around which winds blow counterclockwise in the Northern Hemisphere. Also the term used for a hurricane in the Indian Ocean and in the Western Pacific Ocean.Cyclonic Circulation (or Cyclonic Rotation)- Circulation (or rotation) which is in the same sense as the Earth's rotation, i.e., counterclockwise (in the Northern Hemisphere) as would be seen from above.Dart Leader- In lightning, the leader which, after the first stroke, initiates each succeeding stroke of a composite flash of lightning.Debris Cloud- A rotating "cloud" of dust or debris, near or on the ground, often appearing beneath a condensation funnel and surrounding the base of a tornado.Decouple- The tendency for the surface wind to become much lighter than wind above it at night when the surface temperature cools.Degree Day- a measure of the departure of the daily mean temperature from the normal daily temperature; heating and cooling Degree Days are the departure of the daily mean temperature from sixty-five degrees Fahrenheit. Dendrite- hexagonal ice crystals with complex and often fernlike branches.Dense Fog- a fog in which the visibility is less than one-quarter mile.。

SU(2) Charges as Angular-momentum in N=1 Self-dual Supergravity

a r X i v :h e p -t h /9902063v 1 9 F eb 1999SU(2)Charges as Angular-momentum in N =1Self-dual Supergravity ∗Sze-Shiang Feng ¶§#,Zi-Xing Wang †,Xi-Jun Qiu ¶§CCAST(World Lab.),P.O.Box.8730,Beijing 100080¶Physics Department,Shanghai University,201800,Shanghai,China #e-mail:xjqiu@ #†Institute of Nuclear Research,Academia Sinica,201800,Shanghai,China§Center for String Theory,Shanghai Teacher’s University,200234,Shanghai,ChinaAbstractThe N =1self-dual supergravity has SL (2,C )symmetry .This symme-try results in SU (2)charges as the angular-momentum.As in the non-supersymmetric self-dual gravity,the currents are also of their potentials and are therefore identically conserved.The charges are generally invari-ant and gauge covariant under local SU (2)transforms approaching to be rigid at spatial inﬁnity.The Poisson brackets constitute su (2)algebra and hence can be interpreted as the generally covariant conservative angular-momentum.PACS number(s):11.30.Cp,11.30.Pb,04.20.Me,04.65.+e.Key words:SU (2)charge,angular-momentum,supergravityThe study of self-dual gravities has drawn much attention in the past decade since the discovery of Ashtekar’s new variables,in terms of which the constraints can be greatlysimpliﬁed[1]-[2].The new phase variables consist of densitized SU(2)soldering forms ˜e i A B from which a metric density is obtained according to the deﬁnition q ij=−Tr˜e i˜e j, and a complexiﬁed connection A iA B which carries the momentum dependence in its imag-inary part.The original Ashtekar’s self-dual canonical gravity permits also a Lagrangian formulation[3][4].The supersymmetric extension of this Lagrangian formulation,which is equivalent to the simple real supergravity,was proposed by Jacobson[5],and the cor-responding Ashtekar complex canonical transform was given by Gorobey et al[6].The Lagrangian density is[5]L J=12(e AA′∧e BA′∧F A B+ie AA′∧¯ψA′∧DψA)(1) The dynamical variables are the real tetrad e AA′(the”real”means¯e A′A=e AA′),the traceless left-handed SL(2.C)connection AµMN and the complex anticommuting spin-3ψA∧¯ψA′(4)2provided e AA′is real.The Lagrangian1L J is invariant modulo theﬁeld equations.The(3+1)decomposition is eﬀected asL J=˜e kAB˙A kAB+˜πkA˙ψkA−H(7) H:=e0AA′H AA′+ψ0A S A+ˆS A′¯ψ0A′+A0AB J AB+(total divergence)(8) The canonical momenta are˜e kAB:=−12ǫijk e i AA′e B j A′(9)˜πkA:=i2ǫijk e i AA′¯ψjA′(10)and the constraints areH AA′:=12ǫijk(e i BA′F jkB A−i¯ψi A′D jψk A)(11)S A:=D k˜πkA(12)ˆS A′:=i2ǫijk e i AA′D jψkA(13)J AB:=D k˜e kAB−˜πk(Aψk B)(14) The0-components e0AA′,ψ0A,¯ψ0A′and A0AB are just the Lagrange multipliers and the dynamical conjugate pairs are(˜e kAB,A jAB),(˜πkA,ψkA).The constraints H AA′=0and ˆS A′=0generate the following two¨H AB:=(˜e j˜e k F jk)AB+2˜πj˜e k D[jψk]ǫAB+2(˜πj D[jψk])˜e kAB=0(15)S†A:=12ǫijk˜e i AB D jψkB=0(16) The equations of motion will be properly expressed in Hamiltonian form˙f={H,f}if we assign the Poisson brackets{˜e kAB(x),A jAB(y}=δj kδ(M AδN)Bδ3(x,y)(17){˜πkA(x),ψjA(y)}=−δj kδM Aδ3(x,y)(18) all other brackets among these quantities being zero.This is the outline of the theory.In our previous works,we have obtained the SU(2)charges and the energy-momentumin the Ashtekar’s formulation of Einstein gravity[7]-[8]and they are closely related to the angular-momentum[9]-[11]and the energy-momentum[12]in the vierbein formalism of Einstein gravity.The fact that the algebra formed by their Poisson brackets do constitute the3-Poincare algebra on the Cauchy surface supports from another aspect that their deﬁnitions are reasonable.Similarly,the study of SU(2)charges in the self-dual super-gravity considered is also an interesting subject.In the following,we will employ the SL(2,C)invariance to obtain the conservative charges as we did previously[8]Under any SL(2,C)transformeµAA′→L A B¯R A′B′eµBB′,ψA→L A BψB,¯ψA′→¯R A′B′¯ψB′AµMN→L M A AµA B(L−1)BN+L M A∂µ(L−1)AN(19) L J is invariant.L and¯R may not neccessarily related by complex conjugation.Note that L AB=−(L−1)BA,the transform of A may also be written asAµMN→L M A L N B AµAB−L M A∂µL NA(20) For inﬁnitesimal transform,L A B=δA B+ξA B whereξAB=−ξBA are inﬁnitesimal parametres.Thus we haveδξA=[ξ,A]−dξ,δψ=ξψ(21)When calculating the variation of the Lagrangian,one must take into consideration of the anticommuting feature of the gravitinoﬁeld.We write the variation in the way thatδL J=δφA(∂∂∂µφA)L J+∂µ(δφA∂∂φAand∂∂∂ρAσA B+δψσA∂L J√√we have therefore the conservation of SU(2)charges∂µ˜jµAB=0(25) where˜jρAB =12ǫµνρσ(eµA A′eνMA′AσB M−eµMA′eνBA′AσMA+i2eµB A′¯ψνA′ψσA)(26)ThusJ AB= Σ˜j0AB d3x(27) where˜j0 AB =12ǫijk(e iA A′e jMA′A kB M−e i MA′e jBA′A kMA+i2e iB A′¯ψjA′ψkA)(28)Using eq(9)and eq(10),˜j0AB can be written as˜j0AB=[˜e k,A k]AB+˜πk(Aψk B)(29) The constraint J AB=0guarantees thatJ AB≈Σ∂k˜e k AB= ∂Σ˜e k AB ds i(30)where ds i=12ψB)∧¯ψA′)=0,we haveǫρµνσ[eµA A′(∂σeνBA′+AσB M eνMA′+i2¯ψνA′ψσA)]=0(31)so˜jρAB =−12ǫρµνσ∂σ(eµA A′eνBA′)(32)Usinge[µA A′eν]BA′=e[µAC eν]B C−i√√√2n[i e j]AB)=12ǫijk∂k(e i e j)AB=∂k˜e k AB(34)which is exactly the same as eq.(30)We can thus have the Poisson brackets{J AB,J MN}={∂Σ˜e k AB ds k, Σ(˜e i M P A iP N+˜e i N P A iP N)d3x}=1√2E l we have{J m,J n}=ǫmnl J l(37) Therefore the su(2)algebra is restored.As in the non-supersymmetric case[8],we can also obtain only the SU(2)charges instead of the whole SL(2,C)charges.Yet,the angualr-momentum J ab obtained in[9]-[10]is completely contained in J MN since we have from eq(32)that˜jρAB =−12ǫρσµν∂σ(eµa eνb)(39)and the angualr-momentum isJ ab= Σ˜j0ab d3x(40) HenceJ MN=−12(J ij E[iAC E j]BC−i√√2ǫijk J jk are the spatial rotations and K i=J0i=−J0i are the Lorentz boosts. ThereforeJ i=1Bear in mind that both12(L i+iK i)obey the su(2)algbra[13].Actually, the boost charges are vanishing as can be seen from eq(30).Thus we can obtain the angualr-momentum,in the self-dual simple supergravity once J MN is known.We make a few remarksﬁnally.The total charges take the same integral form as those in the non-supersymmetric case.Though we can obtain the SU(2)sector of the SL(2,C) charges,the information of the angular-momentum is completely contained in the SU(2) charges.It can be seen from the surface integrals that the angular-momentum is governed by the r−2part of˜e i.As in[1]-[2],we always assume that the phase space variables are subject to the boundary conditions.eµAB|∂Σ=(1+M(θ,φ)References[1]A Ashtekar Phys.Rev.Lett.57(1986):2244;Phys.Rev.D36(1987):1587.[2]A.Ashtekar New Perspectives in Canonical Gravity(Lecture Notes,1988,Naples:Biblipolis).[3]T.Jacobson&L.Smolin Phys.Lett B196(1987):39.[4]J.Samuel Pramana J Phys.28(1987):L429.[5]T.Jacobson Class.Quan.Grav.5(1988):923.[6]N.N.Gorobey&A.S.Lukyanenko Class.Quan.Grav.7(1990):67.[7]S.S.Feng&Y.S.Duan Gen.Rel.Grav.27(8)(1995):887.[8]S.S.Feng&Y.S.Duan Commu.Theor.Phys.25(1996):485.[9]Y.S.Duan&S.S.Feng Commu.Theor.Phys.25(1996):99.[10]S.S.Feng&H.S.Zong Inter.J.Theor.Phys.35(1996):267.S.S.Feng&Y.S.DuanGrav.&Cos.1(1995):319.[11]S.S.Feng Nucl.Phys.B468(1996):163.[12]Y.S.Duan&J.Y.Zhang Acta.Phys.Sini19(1963):589.[13]S.Weinberg The Quantum Theory of Field Vol.I(Cambridge University Press,1995)[14]Y.B.Dai Gauge Theory of Interactions in Chinese(Science Publishers1987).。

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a r X i v :c o n d -m a t /9606152v 1 21 J u n 1996typeset using JPSJ.sty <ver.0.8>Origin of Magic Angular Momentum in a Quantum Dot under StrongMagnetic FieldTaku Seki ,∗Yoshio Kuramoto ∗∗and Tomotoshi NishinoDepartment of Physics,Tohoku University,Sendai 980-77(Received )This paper investigates origin of the extra stability associated with particular values (magic numbers)of the total angular momentum of electrons in a quantum dot under strong magnetic ﬁeld.The ground-state energy,distribution functions of density and angular momentum,and pair correlation function are calculated in the strong ﬁeld limit by numerical diagonalization of the system containing up to seven electrons.It is shown that the composite fermion picture explains the small magic numbers well,while a simple geometrical picture does better as the magic number bination of these two pictures leads to identiﬁcation of all the magic numbers.Relation of the magic-number states to the Wigner crystal and the fractional quantum Hall state is discussed.KEYWORDS:quantum dot,fractional quantum Hall state,composite fermion,exact diagonalization,magic number,Wigner crystal §1.Introduction The two-dimensional electron system with a small number N of electrons is realized as the quantum dot formed at a semiconductor interface.1)Correspondingly there is a growing theoreticalinterest in the system.It has been found 2,3,4,5,6,7,8,9,10)that if a system is placed in a strong magnetic ﬁeld,extra stability arises for a special set of angular momentum M.These values of M are called magic numbers.According to numerical diagonalization,the magic number states occur with an interval ∆M =N for systems with N ≤5,and that these states have a polygonal pattern with N apexes in the conﬁguration of electrons.5,7)The magic number states prevail the ground-state phase diagram in the plane of magnetic ﬁeld vs the conﬁning potential for N =5and 6.8)On the other hand,it has been shown 6,9)that many of the magic number states can be explained in terms of the composite fermion (CF)picture 11)for the fractional quantum Hall (FQH)states.The overlap is close to unity 9,12)between a trial wave function based on the CF theory andthe exact ground state wave function in the presence of the Coulomb interaction.However,some magic number states do notﬁt into the interpretation of the CF theory.The purpose of this paper is to clarify the origin of the magic numbers on the basis of exactly derived results for the energy,and one-and two-body distribution functions.In addition to the previously known series with∆M=N,weﬁnd another series with∆M=N−1of the magic number states for N≥6.We provide simple geometrical interpretation on the origin of new series of magic bining with previous interpretations,we come to uniﬁed understanding of all the magic numbers.The paper is organized as follows:In§2we deﬁne the model which involves truncation to the lowest Landau level.Section3presents numerical results on the ground state energy as a function of the total angular momentum.The magic numbers are identiﬁed for systems with6and7electrons. The results on one-and two-body distribution functions are shown in§4where the characteristic features of the magic-number states are made explicit.We provide in§5a geometrical interpretation of the magic number states,and discuss the relation to the CF picture.9)In§6we combine the geometrical and CF interpretations,and point out the range of validity for each interpretation. This reveals signiﬁcance of magic-number states as showing crossover from an incipient fractional quantum Hall state to an incipient Wigner crystal.§2.Model and Choice of BasisWe consider a model for a quantum dot in two dimensions with the HamiltonianH=H0+V1+V2.(2.1) Here H0,V1,and V2are given byH0=1i∇i+e2m∗ω2Ni=1|z i|2,V2=e2|zi−z j|,(2.3)withωbeing the frequency of the harmonic oscillator,and z=x+iy.Other notations are the standard ones.The vector potential is chosen to be A i=(y i B/2,−x i B/2).We assume that the magneticﬁeld B is so strong that all electrons are in the lowest Landau level(LLL),with spins being completely polarized.One-electron state is speciﬁed by its orbital angular momentum only. We ignore the spin degrees of freedom and mixing between the LLL and higher Landau levels. Hence the Zeeman term has been dropped in our model.The normalized single electron wave function in the LLL is given byφm(z)=12πℓ22m m!z m exp(−|z|2where m is the angular momentum and ℓthe magnetic length deﬁned by ℓ=(¯h c/eB )12 m i <m 1m 2|V 2|m 3m 4>a †m 1a †m 2a m 3a m 4,(2.5)where a m is the annihilation operator of an electron with m .The matrix elements are given by<m |H 0|m >=12m ∗ω2ℓ2(m +1),(2.7)and the Coulomb matrix element is computed numerically from<m 1m 2|V 2|m 3m 4>=(e 2 Γ(d +1)2× ∞0d x x 2d 1F 1(j −k +1;d +1;−x 22).Here 1F 1is Kummer’s hypergeometric function,and m 1=j −k,m 2=k,m 3=k +d,m 4=j −k −d with d ≥0.We discard in the following the constant kinetic term <m |H 0|m >=12e 2/(ε0ℓ).(2.8)The length is scaled by ℓ,and energy is scaled by e 2/(ε0ℓ)from now on.Since the conﬁning potential and the Coulomb interaction have the rotational symmetry,the total angular momentum M can be used to label the many body eigenstates.We introduce the distribution function of angular momentum byn m =<Φ0|a †m a m |Φ0>,(2.9)where |Φ0>is the normalized ground state.Then the total angular momentum is given byM =∞ m =0mn m .(2.10)Next we introduce a ﬁeld operator ψ(z )truncated within the LLL:ψ(z )=m φm (z )a m .(2.11)Then the charge density n (z )is given byn (z )=<Φ0|ψ†(z )ψ(z )|Φ0>,(2.12)and the total number N by its integral over the whole space.Finally the pair (two-body)correlationfunction is deﬁned asn (z,z ′)=<Φ0|ψ†(z )ψ†(z ′)ψ(z ′)ψ(z )|Φ0>.(2.13)§3.Two Series of Magic NumbersBy using the Lanczos method to diagonalize the Hamiltonian numerically,we calculate the lowest energy and corresponding eigenvectors for each M and N.For given N and M,the one body angular momentum takes a value from0to M−N(N−1)/2.The upper limit less than M comes from the Pauli exclusion principle.In constructing the Hamiltonian matrix,we take all the necessary basis for m without truncation.We note the following relation:V1|Φ0>=γ(M+N)|Φ0>,(3.1) which follows from eqs.(2.7).Thus the wave function|Φ0>is actually independent ofγ,and the numerical result forγ=0is suﬃcient to derive the energy for other cases ofγ.Figure1(a)shows the results for the ground-state energy vs M in the case of N=6.A numberFig.1.The ground-state energy vs the total angular momentum in the case of(a)N=6and(b)N=7.The parameterγrepresents the strength of the harmonic conﬁnement as explained in the text.of downward cusps appear clearly.Upon closer inspection oneﬁnds that a part of the cusps with M=15,21,27,33,39,45are represented by the formula1M=In Fig.1(b)we show the ground-state energy in the case of N=7.Weﬁnd clear downward cusps at the series˜S N.This includes M=21,33,39,45,51,57,63and69.However,the number M=27 belonging to˜S N is not a magic number.On the other hand,the cusps corresponding to the series S N are not clear except for M=28.This is in strong contrast with the case of N≤6.We note that a magic number state with M=36is seen in Fig.1(b).This state belongs to neither the series S N nor˜S N.As we discuss later,the CF picture can interpret the magic number M=36.§4.Distribution Functions4.1Angular momentumIn order to clarify the electronic property associated with the magic-number states,we calculate the distribution function n m of the angular momentum.In this section,we report mainly on results for N=6.Figure2shows representative results.As M increases,two types of distribution appearFig.2.One-body angular momentum distribution function for(a)M=33and35and(b)M=39and45.The distribution with double peaks as for M=35and45is called the type-D,while the one with a single peak as for M=33and39is called the type-S.by turns in the magic-number states.One is the distribution with double peaks at the origin as well as at aﬁnite angular momentum.We call the double-peaked distribution the type-D hereafter. This is the case with M=35and45.The other type is the distribution with a single peak which we call the type-S,as for M=33and39.An important observation is that the magic number states with the type-D distribution are all in the series˜S N,while those with the type-S distribution are all in the series S N.These features are common to cases other than N=6.It is instructive to interpret the result in terms of the Laughlin wave functionΨL(z1,...,z N)for ﬁnite N.It is given for general N byΨL(z1,...,z N)= i<j(z i−z j)p exp(−1p=3and N=6,in particular,we obtain m max=15and the total angular momentum M is given by M=3N(N−1)/2=45.We note that the numerically obtained results for M=45in Fig.2(b) has the maximum15of the signiﬁcantly occupied angular momentum in good correspondence to the Laughlin wave function with p=3.4.2Charge densityThe larger angular momentum corresponds to the wave function more extended from the origin. More explicitly we have the relation<m|r2|m>=2ℓ2(m+1).(4.2) We can therefore expect that n(z)of the type-D distribution should have double peaks both near the origin and near the edge of the dot,and that of the type-S distribution should have only a single peak near the edge.We have actually calculated the charge densities for these states with N=6and7.Since the result n(z)depends only on r=|z|,the density is written as n(r)in the following.Figure3shows some exemplary results.It is found that the state with the type-S distributionFig.3.The charge density n(r)vs the distance r from the center of the dot in the case of M=33,35and45.(M=33)indeed has a single peak near the edge of the quantum dot,and those with the type-D distribution(M=35,45)have double peaks.This feature is common to systems with diﬀerent electron numbers.Since the Laughlin-type state with M=pN(N−1)/2belongs simultaneously to series S N and˜S N,it is natural to expect that the charge distribution is also a superposition of the type-D and the type-S distributions.As a result,the distribution should have double peaks. The result in Fig.3shows that this is indeed the case with M=45and N=6.It is seen from eq.(4.2)that the average density of electrons in the quantum dot decreases as the total angular mometum M increases.In§6we use this fact for uniﬁed understanding of magic numbers.4.3Pair correlation functionIn order to see the correlated motion of electrons more closely we calculate the pair correlation function n(z,z′)exactly.This quantity gives us a quantum analogue of a snapshot picture of electrons.More precisely,it gives the distribution of electrons on condition that one of the electrons is nailed down at z′.The pair correlation function for N=5or less has been calculated by Maksym,7)who found a pattern corresponding to the polygon with N apexes for the series S N. For N=6or more,however,we are not aware of exact results reported so far.Moreover nothing is known about the pair distribution for the series˜S N.We deﬁne r0as the peak position of the charge density,which in the case of type-D distribution is to be taken at the position of the outer peak.Then we calculate n(z,z′)numerically,setting|z′|=r0.From Fig.4with N=6,it is clear that electrons are arranged like a hexagon in the magic number state(M=33)belonging to the series S N,and like a pentagon in the magic number state(M=35) of the series˜S N.We recall that a Laughlin-type state withν=1/p belongs simultaneously to theFig.4.Contour map of the pair correlation function n(z,z′)for(a)M=33and(b)M=35.The complex coordinate z′=r0i corresponds to the center of the darkest region with r0=2.199in(a)and r0=2.592in(b). The interval of contours is8.265×10−4in(a)and is6.472×10−4in(b).series S N and˜S N.Then n(z,z′)should be a superposition of pentagon-and hexagon-like patterns. To demonstrate this,the pair correlation function for p=3(M=45)is calculated and is shown in Fig.5.It clearly conﬁrms the expectation.Similar calculation is also performed for N=7.Figure6shows the result for M=39which belongs to the series˜S N.It shows the hexagon-type pattern as expected.On the contrary,we have conﬁrmed that non-magic-number states such as M=35(N=7)and M=34(N=6)do not show a regular polygonal pattern.§5.Interpretation of Magic Numbers5.1Geometrical interpretationAs we have seen in the previous section,all the magic number states with N=6belong either to the series S6or˜S6.In this section,we discuss the relation between the total angular momentum andFig.5.Contour map of the pair correlation function n(z,z′)for M=45with z′=2.915i.The interval of contours is5.045×10−4.Fig.6.Contour map of the pair correlation function n(z,z′)for M=39,N=7with z′=2.442i.The interval of contours is1.343×10−3.the symmetry of the many-electron wave function.The argument extends the previous one which explains the series S N.13)Let us consider the wave functionΦ(z1,z2,···,z N)in the case where the coordinates correspond to apexes of a regular polygon:z n=R exp(inφ)with n=1,2,···,N and φ=2π/N.By applying exp(iˆMφ)whereˆM is the total angular momentum operator,we obtainexp(iˆMφ)Φ(z1,z2,···,z N)=Φ(z N,z1,···,z N−1)=(−1)N−1Φ(z1,z2,···,z N).The second equality follows from the antisymmetry of the fermion wave function.Then we get[exp(iMφ)+(−1)N]Φ(z1,z2,···,z N)=0.(5.1)In order for the regular polygonal pattern with N apexes to be realized,Φ(z1,z2,···,z N)must be nonzero.This gives a selection rule on M as follows:1M=N(j+if N is odd.The Pauli principle in addition requires M≥N(N−1)/2.This selection rule leads to the series S N.In order to examine the case of the polygonal pattern with N−1apexes for N electrons,we put z N=0,z n=R exp(inφ′)with n=1,2,···,N−1andφ′=2π/(N−1).Then we obtain2πM[exp(i),(j=0,1,···)(5.6)2if N is odd.This selection rule leads to the series˜S N.These rules nicely explain the occurrence of the magic number states in the case of N≤6. However for larger N,the conﬁguration with N-apex pattern costs more energy than another apex pattern with one or more electrons in the interior.This is the reason why the magic number series S7is hardly seen in Fig.1(b).The exceptional appearance of S7is the magic number state M=28 with N=7.The stability of the state,however,is better explained in terms of the CF picture as discussed below.Now we turn attention to theν=1/3Laughlin-type state which belongs to the series S N and˜SN simultaneously.Since this state has superposition of the polygonal pattern with N apexes and the one with N−1apexes,an extra stability is expected due to their resonant energy.5.2Composite fermion pictureIn ref.9an interpretation of magic numbers in terms of the CF picture is presented.In this picture the wave function is constructed in the following form:ΨCF(z1,...,z N)= i<j(z i−z j)2q PΨ0(z1,...,z N),(5.7) whereΨ0(z1,...,z N)is a wave function of free electrons,and q is a natural number.The Jastrow-type factor in eq.(5.7)is interpreted as binding a magneticﬂux with strength2q to each electron, hence the name of the CF.11)AlthoughΨ0(z1,...,z N)is not restricted to the LLL,the projection operator P picks out only such component that belongs to the LLL.Thus the total angular mo-mentum M ofΨCF(z1,...,z N)is the sum of the part M0associated withΨ0(z1,...,z N)and that coming from the Jastrow-type ly we obtainM=qN(N−1)+M0.(5.8) According to ref.9the magic number corresponds to suchΨ0(z1,...,z N)that has compact occu-pation of each Landau level from the lowest possible angular momentum.In the case of N≤5theCF picture explains all the magic numbers.9)The CF magic numbers for N=5are the almost the same as the series S5with additional ones M=18and22from the series˜S5.For N=4all magic numbers in S4are also given by the CF picture,while the CF picture gives M=12as the only magic number in the series˜S4.The series S N with the electron number N≤5have been studied by other authors4,5,7)as well.For the case of N=6and q=1,the state with M=45has six electrons in the LLL.This state has M0=15.The other extreme case is the one where each of the six lowest Landau levels has only one electron,which we write as(1,1,1,1,1,1),resulting in M0=−15.If one excludes an occupation where a higher LL has more electrons than a lower one,the number of compact states is counted as11for N=6.However,both occupations(3,1,1,1)and(2,2,2)give M=27,and both(4,1,1)and(3,3)give M=33.The set of the magic numbers with q=1are thenM=15,21,25,27,30,33,35,39,45.All these states belong to either S6or˜S6.With q=2the lowest magic number is M=45and the second lowest is51.A nice feature of the CF picture is to reject M=20as a magic number in accordance with the numerical result in Fig.1(a).On the other hand,the numerical result shows clear cusps at M=40and50in contradiction to the CF picture.Similar analysis is carried out for N=7.The number of compact states is15for each q and the magic numbers with q=1are given byM=21,28,33,35,36,39,41,42,43,45,48,49,51,56,63.By comparing with Fig.2(b)we see that many of them correspond to cusps in the ground state energy.However,the numerical result shows clear cusps also at M=57and69both of which do not correspond to compact states.As N increases,the magic numbers predicted in the CF picture appear more densely than given in the series S N and˜S N.Some of them appear clearly in the numerical result,while others have almost no cusp.Unfortunately the relative stability among magic number states is not given in the CF picture.We remark that among the magic numbers which are not explained in the CF picture,M=40for N=6and M=57for N=7are both next to smaller magic numbers,39and56.§6.Discussion and ConclusionWe have seen that there are apparently conﬂicting interpretations of magic number states:the geometrical one which emphasizes the real space conﬁguration,and the CF one which emphasizes the compact occupation of the one-body angular momentum.In the case of small M,the CF picture works better in rejecting M=20with N=6,or M=27with N=7.However,in the case of large M,the geometrical interpretation works better in explaining M=40and50with N=6, and M=57and69with N=7.In the intermediate values of M,many of the magic numbers arecommon to both the geometrical and the CF pictures.By combining both pictures we can account for all the magic numbers.It should be noted that the smaller M corresponds to higher density of electrons in the macro-scopic limit.Thus the crossover in the eﬀectiveness of the geometrical and the CF interpretations seems to reﬂect the transition from a FQH liquid at high density to the Wigner solid at low density in the macroscopic limit.The success of both interpretations in the intermediate range of M is intriguing.This suggests a smooth crossover forﬁnite N from the incipient FQH liquid to the incipient Wigner solid with decreasing density.In this connection we remind the early work14)which argues that particular densities for the stable FQH states also stabilize the Wigner crystal with large zero-point motion and the ring exchange. It is thus not surprising that some of the electronic states in the quantum dot connect both to the FQH state and the Wigner crystal.Experimentally,a particular advantage of the quantum dot is the wide range of controllable electron numbers N.Thus one may observe how the electronic states change from the atomic type to the macroscopic type as N is increased.This feasibility is in strong contrast to genuine atoms.In summary,we have investigated the quantum dot system in a strong magneticﬁeld by numerical diagonalization of the Hamiltonian.We have provided uniﬁed understanding of all the magic numbers in the total angular momentum.AcknowledgmentThe authors are grateful to Y.Kato,T.Nihonyanagi,S.Tokizaki and H.Yokoyama for fruitful discussions.The numerical calculations are performed by SX3-44R at the computing center of Tohoku University.This research was supported by a Grant-in-Aid for Scientiﬁc Research on Priority Area from the Ministry of Education,Science,Sports and Culture.。