physics assignment 4th semester

8/6/2019 Physics Assignment 4th Semester

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G.C. University, Faisalabad

Topic: Angular Momentum

Submitted to:

Mam Areeb FatimaSubmitted by:

Muzaffar HussainRoll no : 1388

Subject : Physics

Department : Computer Sciences

_______________________


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Angular momentum:

Definition

The angular momentum L of a particle about a given origin is defined as:

where r is the position vector of the particle relative to the origin, p is the linear momentum of the

particle, and denotes the cross product.
http://en.wikipedia.org/wiki/Momentumhttp://en.wikipedia.org/wiki/Cross_producthttp://en.wikipedia.org/wiki/Momentumhttp://en.wikipedia.org/wiki/Cross_product


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As seen from the definition, the derived SI units of angular momentum are newtonmetre seconds

(Nms or kgm2s1) orjoule seconds. Because of the cross product, L is a pseudovector

perpendicular to both the radial vectorr and the momentum vectorp and it is assigned a sign bythe right-hand rule.

For an object with a fixed mass that is rotating about a fixed symmetry axis, the angularmomentum is expressed as the product of the moment of inertia of the object and its angular

velocity vector:

whereIis the moment of inertia of the object (in general, a tensorquantity), and is the angular

velocity.

The angular momentum of a particle or rigid body in rectilinear motion (pure translation) is avector with constant magnitude and direction. If the path of the particle or rigid body passes

through the given origin, its angular momentum is zero.

Angular momentum is also known as moment ofmomentum.

Angular momentum of a collection of particles

If a system consists of several particles, the total angular momentum about a point can beobtained by adding (or integrating) all the angular momenta of the constituent particles.

Angular momentum simplified using the center of mass

It is very often convenient to consider the angular momentum of a collection of particles abouttheircenter of mass, since this simplifies the mathematics considerably. The angular momentum

of a collection of particles is the sum of the angular momentum of each particle:

where Ri is the position vector of particle i from the reference point, mi is its mass, and Vi is its

velocity. The center of mass is defined by:

where the total mass of all particles is given by

It follows that the velocity of the center of mass is
http://en.wikipedia.org/wiki/Derived_SI_unithttp://en.wikipedia.org/wiki/Newton_(unit)http://en.wikipedia.org/wiki/Metrehttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Joulehttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Pseudovectorhttp://en.wikipedia.org/wiki/Right-hand_rulehttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Moment_(physics)http://en.wikipedia.org/wiki/Momentumhttp://en.wikipedia.org/wiki/Center_of_masshttp://en.wikipedia.org/wiki/Derived_SI_unithttp://en.wikipedia.org/wiki/Newton_(unit)http://en.wikipedia.org/wiki/Metrehttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Joulehttp://en.wikipedia.org/wiki/Secondhttp://en.wikipedia.org/wiki/Pseudovectorhttp://en.wikipedia.org/wiki/Right-hand_rulehttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Moment_(physics)http://en.wikipedia.org/wiki/Momentumhttp://en.wikipedia.org/wiki/Center_of_mass


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If we define ri as the displacement of particle i from the center of mass, and vi as the velocity ofparticle i with respect to the center of mass, then we have

and

and also

and

so that the total angular momentum with respect to the center is

The first term is just the angular momentum of the center of mass. It is the same angular

momentum one would obtain if there were just one particle of mass M moving at velocity V

located at the center of mass. The second term is the angular momentum that is the result of theparticles moving relative to their center of mass. This second term can be even further simplified

if the particles form a rigid body, in which case it is the product ofmoment of inertia and angular

velocity of the spinning motion (as above). The same result is true if the discrete point massesdiscussed above are replaced by a continuous distribution of matter.

Fixed axis of rotation

For many applications where one is only concerned about rotation around one axis, it is sufficient

to discard the pseudovector nature of angular momentum, and treat it like a scalar where it is

positive when it corresponds to a counter-clockwise rotation, and negative clockwise. To do this,just take the definition of the cross product and discard the unit vector, so that angular momentum

becomes:

where r,p is the angle between r and p measured from r to p; an important distinction because

without it, the sign of the cross product would be meaningless. From the above, it is possible toreformulate the definition to either of the following:

where is called theleverarm distance to p.
http://en.wikipedia.org/wiki/Rigid_bodyhttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Leverhttp://en.wikipedia.org/wiki/Leverhttp://en.wikipedia.org/wiki/Rigid_bodyhttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Lever


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The easiest way to conceptualize this is to consider the lever arm distance to be the distance from

the origin to the line that p travels along. With this definition, it is necessary to consider the

direction ofp (pointed clockwise or counter-clockwise) to figure out the sign of L. Equivalently:

where is the component ofp that is perpendicular to r. As above, the sign is decided based on

the sense of rotation.

For an object with a fixed mass that is rotating about a fixed symmetry axis, the angularmomentum is expressed as the product of the moment of inertia of the object and its angular

velocity vector:

whereIis the moment of inertia of the object (in general, a tensorquantity) and is the angularvelocity.

It is a misconception that angular momentum is always about the same axis as angular velocity.

Sometime this may not be possible, in these cases the angular momentum component along the

axis of rotation is the product of angular velocity and moment of inertia about the given axis ofrotation.

As the kinetic energyKof a massive rotating body is given by

K=I2 / 2

it is proportional to the square of the angular velocity.

Conservation of angular momentum
http://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Kinetic_energyhttp://en.wikipedia.org/wiki/File:PrecessionOfATop.svghttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Moment_of_inertiahttp://en.wikipedia.org/wiki/Tensorhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Angular_velocityhttp://en.wikipedia.org/wiki/Kinetic_energy


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The torque caused by the two opposing forces Fg and -Fg causes a change in the angular

momentum L in the direction of that torque (since torque is the time derivative of angular

momentum). This causes the top toprecess.

In a closed system angular momentum is constant. This conservation law mathematically follows

from continuous directional symmetry of space (no direction in space is any different from anyother direction). SeeNoether's theorem

The time derivative of angular momentum is called torque:

(The cross-product of velocity and momentum is zero, because these vectors are parallel.) So

requiring the system to be "closed" here is mathematically equivalent to zero external torqueacting on the system:

where is any torque applied to the system of particles. It is assumed that internal interaction

forces obey Newton's third law of motion in its strong form, that is, that the forces betweenparticles are equal and opposite and act along the line between the particles.

In orbits, the angular momentum is distributed between the spin of the planet itself and the

angular momentum of its orbit:

;

If a planet is found to rotate slower than expected, then astronomers suspect that the planet isaccompanied by a satellite, because the total angular momentum is shared between the planet and

its satellite in order to be conserved.

The conservation of angular momentum is used extensively in analyzing what is called centralforce motion. If the net force on some body is directed always toward some fixed point, thecenter, then there is no torque on the body with respect to the center, and so the angular

momentum of the body about the center is constant. Constant angular momentum is extremely

useful when dealing with the orbits ofplanets and satellites, and also when analyzing the Bohr

model of the atom.

The conservation of angular momentum explains the angular acceleration of an ice skater as she

brings her arms and legs close to the vertical axis of rotation. By bringing part of mass of her

body closer to the axis she decreases her body's moment of inertia. Because angular momentum isconstant in the absence of external torques, the angular velocity (rotational speed) of the skater

has to increase.
http://en.wikipedia.org/wiki/Torquehttp://en.wikipedia.org/wiki/Precesshttp://en.wikipedia.org/wiki/Noether's_theoremhttp://en.wikipedia.org/wiki/Torquehttp://en.wikipedia.org/wiki/Newton's_laws_of_motionhttp://en.wikipedia.org/wiki/Orbithttp://en.wikipedia.org/wiki/Planethttp://en.wikipedia.org/wiki/Satellitehttp://en.wikipedia.org/wiki/Bohr_modelhttp://en.wikipedia.org/wiki/Bohr_modelhttp://en.wikipedia.org/wiki/Atomhttp://en.wikipedia.org/wiki/Torquehttp://en.wikipedia.org/wiki/Precesshttp://en.wikipedia.org/wiki/Noether's_theoremhttp://en.wikipedia.org/wiki/Torquehttp://en.wikipedia.org/wiki/Newton's_laws_of_motionhttp://en.wikipedia.org/wiki/Orbithttp://en.wikipedia.org/wiki/Planethttp://en.wikipedia.org/wiki/Satellitehttp://en.wikipedia.org/wiki/Bohr_modelhttp://en.wikipedia.org/wiki/Bohr_modelhttp://en.wikipedia.org/wiki/Atom


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The same phenomenon results in extremely fast spin of compact stars (like white dwarfs, neutron

stars andblack holes) when they are formed out of much larger and slower rotating stars (indeed,

decreasing the size of object 104 times results in increase of its angular velocity by the factor 108).

The conservation of angular momentum in EarthMoon system results in the transfer of angular

momentum from Earth to Moon (due to tidal torque the Moon exerts on the Earth). This in turnresults in the slowing down of the rotation rate of Earth (at about 42 nsec/day), and in gradual

increase of the radius of Moon's orbit (at ~4.5 cm/year rate).

Angular momentum in relativistic mechanics

In modern (late 20th century) theoretical physics, angular momentum is described using a

different formalism. Under this formalism, angular momentum is the 2-formNoether charge

associated with rotational invariance (As a result, angular momentum is not conserved for generalcurved spacetimes, unless it happens to be asymptotically rotationally invariant). For a system of

point particles without any intrinsic angular momentum, it turns out to be

(Here, the wedge product is used.).

In the language of four-vectors and tensors the angular momentum of a particle in relativistic

mechanics is expressed as a antisymmetric tensor of second order

Angular momentum in quantum mechanics

In quantum mechanics, angular momentum is quantized that is, it cannot vary continuously, but

only in "quantum leaps" between certain allowed values. The orbital angular momentum of asubatomic particle, that is due to its motion through space, is always a whole-number multiple of

("h-bar," known as the reduced Planck's constant). Furthermore, experiments show that most

subatomic particles have a permanent, built-in angular momentum, which is not due to their

motion through space. This spin angular momentum comes in units of . For example, an

electron standing at rest has an angular momentum of .

Basic definition

The classical definition of angular momentum as depends on six numbers: rx, ry, rz,px, py, and pz. Translating this into quantum-mechanical terms, the Heisenberg uncertaintyprinciple tells us that it is not possible for all six of these numbers to be measured simultaneously

with arbitrary precision. Therefore, there are limits to what can be known or measured about aparticle's angular momentum. It turns out that the best that one can do is to simultaneously

measure both the angular momentum vector's magnitude and its component along one axis.
http://en.wikipedia.org/wiki/White_dwarfhttp://en.wikipedia.org/wiki/Neutron_starhttp://en.wikipedia.org/wiki/Neutron_starhttp://en.wikipedia.org/wiki/Black_holehttp://en.wikipedia.org/wiki/2-formhttp://en.wikipedia.org/wiki/Noether_chargehttp://en.wikipedia.org/wiki/Wedge_producthttp://en.wikipedia.org/wiki/Four-vectorhttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantization_(physics)http://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Reduced_Planck's_constanthttp://en.wikipedia.org/wiki/Spin_(physics)http://en.wikipedia.org/wiki/Uncertainty_principlehttp://en.wikipedia.org/wiki/Uncertainty_principlehttp://en.wikipedia.org/wiki/White_dwarfhttp://en.wikipedia.org/wiki/Neutron_starhttp://en.wikipedia.org/wiki/Neutron_starhttp://en.wikipedia.org/wiki/Black_holehttp://en.wikipedia.org/wiki/2-formhttp://en.wikipedia.org/wiki/Noether_chargehttp://en.wikipedia.org/wiki/Wedge_producthttp://en.wikipedia.org/wiki/Four-vectorhttp://en.wikipedia.org/wiki/Quantum_mechanicshttp://en.wikipedia.org/wiki/Quantization_(physics)http://en.wikipedia.org/wiki/Quantum_statehttp://en.wikipedia.org/wiki/Reduced_Planck's_constanthttp://en.wikipedia.org/wiki/Spin_(physics)http://en.wikipedia.org/wiki/Uncertainty_principlehttp://en.wikipedia.org/wiki/Uncertainty_principle


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Mathematically, angular momentum in quantum mechanics is defined like momentum - not as a

quantity but as an operatoron the wave function:

where r and p are the position and momentum operators respectively. In particular, for a singleparticle with no electric charge and no spin, the angular momentum operatorcan be written in the

position basis as

where is the vector differential operator del (also called "Nabla"). This orbital angular

momentum operatoris the most commonly encountered form of the angular momentum operator,

though not the only one. It satisfies the following canonical commutation relations:

,

where

lmn is the (antisymmetric) Levi-Civita symbol,

[X,Y] =XY YXis the commutator.

From this follows

Since,

it follows, for example,
http://en.wikipedia.org/wiki/Momentum#Momentum_in_quantum_mechanicshttp://en.wikipedia.org/wiki/Operator_(physics)http://en.wikipedia.org/wiki/Wave_functionhttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Spin_(physics)http://en.wikipedia.org/wiki/Angular_momentum_operatorhttp://en.wikipedia.org/wiki/Differential_operatorhttp://en.wikipedia.org/wiki/Delhttp://en.wikipedia.org/wiki/Nabla_symbolhttp://en.wikipedia.org/wiki/Canonical_commutation_relationhttp://en.wikipedia.org/wiki/Levi-Civita_symbolhttp://en.wikipedia.org/wiki/Commutatorhttp://en.wikipedia.org/wiki/Momentum#Momentum_in_quantum_mechanicshttp://en.wikipedia.org/wiki/Operator_(physics)http://en.wikipedia.org/wiki/Wave_functionhttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Spin_(physics)http://en.wikipedia.org/wiki/Angular_momentum_operatorhttp://en.wikipedia.org/wiki/Differential_operatorhttp://en.wikipedia.org/wiki/Delhttp://en.wikipedia.org/wiki/Nabla_symbolhttp://en.wikipedia.org/wiki/Canonical_commutation_relationhttp://en.wikipedia.org/wiki/Levi-Civita_symbolhttp://en.wikipedia.org/wiki/Commutator


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Addition of quantized angular momenta

For more details on this topic, see Clebsch-Gordan coefficients.

Given a quantized total angular momentum which is the sum of two individual quantized

angular momenta and ,

the quantum numberj associated with its magnitude can range from | l1 l2 | to l1 + l2 in integer

steps where l1 and l2 are quantum numbers corresponding to the magnitudes of the individualangular momenta.

Angular momentum as a generator of rotations

If is the angle around a specific axis, for example the azimuthal angle around the z axis, thenthe angular momentum along this axis is the generatorof rotations around this axis:

The eigenfunctions of Lz are therefore , and since has a period of 2, ml must be aninteger.

For a particle with a spinS, this takes into account only the angular dependence of the location ofthe particle, for example its orbit in an atom. It is therefore known as orbital angular momentum.

However, when one rotates the system, one also changes the spin. Therefore the total angular

momentum, which is the full generatorof rotations, isJi =Li + Si Being an angular momentum,J satisfies the same commutation relations as L, as will be explained below, namely

from which follows

Relation to spherical harmonics

Angular momentum operators usually occur when solving a problem with spherical symmetry in

spherical coordinates. Then, the angular momentum in space representation is:
http://en.wikipedia.org/wiki/Clebsch-Gordan_coefficientshttp://en.wikipedia.org/wiki/Quantum_numberhttp://en.wikipedia.org/wiki/Noether's_theoremhttp://en.wikipedia.org/wiki/Eigenfunctionhttp://en.wikipedia.org/wiki/Spin_(physics)http://en.wikipedia.org/wiki/Orbital_angular_momentumhttp://en.wikipedia.org/wiki/Spin_(physics)http://en.wikipedia.org/wiki/Total_angular_momentumhttp://en.wikipedia.org/wiki/Total_angular_momentumhttp://en.wikipedia.org/wiki/Noether's_theoremhttp://en.wikipedia.org/wiki/Azimuthal_quantum_number#Total_angular_momentum_of_an_electron_in_the_atomhttp://en.wikipedia.org/wiki/Spherical_symmetryhttp://en.wikipedia.org/wiki/Spherical_coordinateshttp://en.wikipedia.org/wiki/Clebsch-Gordan_coefficientshttp://en.wikipedia.org/wiki/Quantum_numberhttp://en.wikipedia.org/wiki/Noether's_theoremhttp://en.wikipedia.org/wiki/Eigenfunctionhttp://en.wikipedia.org/wiki/Spin_(physics)http://en.wikipedia.org/wiki/Orbital_angular_momentumhttp://en.wikipedia.org/wiki/Spin_(physics)http://en.wikipedia.org/wiki/Total_angular_momentumhttp://en.wikipedia.org/wiki/Total_angular_momentumhttp://en.wikipedia.org/wiki/Noether's_theoremhttp://en.wikipedia.org/wiki/Azimuthal_quantum_number#Total_angular_momentum_of_an_electron_in_the_atomhttp://en.wikipedia.org/wiki/Spherical_symmetryhttp://en.wikipedia.org/wiki/Spherical_coordinates


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When solving to find eigenstates of this operator, we obtain the following

where

are the spherical harmonics.

Thus, a particle whose wave function is the spherical harmonic Yl,m has an orbital angularmomentum

with a z-component

Angular momentum in electrodynamics

When describing the motion of a charged particle in the presence of an electromagnetic field, thecanonical momentum p is not gauge invariant. As a consequence, the canonical angular

momentum is not gauge invariant either. Instead, the momentum that is physical, the

so-called kinetic momentum, is

where e is the electric charge, c the speed of light andA the vector potential. Thus, for example,the Hamiltonian of a charged particle of mass m in an electromagnetic field is then

where is the scalar potential. This is the Hamiltonian that gives the Lorentz force law. Thegauge-invariant angular momentum, or "kinetic angular momentum" is given by
http://en.wikipedia.org/wiki/Eigenstatehttp://en.wikipedia.org/wiki/Spherical_harmonichttp://en.wikipedia.org/wiki/Spherical_harmonichttp://en.wikipedia.org/wiki/Electromagnetic_fieldhttp://en.wikipedia.org/wiki/Canonical_momentumhttp://en.wikipedia.org/wiki/Gauge_invarianthttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Speed_of_lighthttp://en.wikipedia.org/wiki/Vector_potentialhttp://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)http://en.wikipedia.org/wiki/Scalar_potentialhttp://en.wikipedia.org/wiki/Lorentz_force_lawhttp://en.wikipedia.org/wiki/Eigenstatehttp://en.wikipedia.org/wiki/Spherical_harmonichttp://en.wikipedia.org/wiki/Spherical_harmonichttp://en.wikipedia.org/wiki/Electromagnetic_fieldhttp://en.wikipedia.org/wiki/Canonical_momentumhttp://en.wikipedia.org/wiki/Gauge_invarianthttp://en.wikipedia.org/wiki/Electric_chargehttp://en.wikipedia.org/wiki/Speed_of_lighthttp://en.wikipedia.org/wiki/Vector_potentialhttp://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)http://en.wikipedia.org/wiki/Scalar_potentialhttp://en.wikipedia.org/wiki/Lorentz_force_law


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physics assignment 4th semester

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