Angular Momentum in Quantum Mechanics

Name: Angular Momentum in Quantum Mechanics
Uploaded: 2024-11-15T20:34:30.000Z
Duration: 58 min 8 s

Angular Momentum and Commutation Relations in Quantum Mechanics

Angular Momentum in 3D Space

A particle's position in 3D space is represented by a vector R . If its momentum is not aligned with R , it can exhibit circular motion, leading to angular momentum defined as L = R times P .

The components of angular momentum are calculated using determinants:

X component: L_x = YP_z - ZP_y

Y component: L_y = ZP_x - XP_z

Z component: L_z = XP_y - YP_x .

Operators in Quantum Mechanics

In quantum mechanics, position and momentum are treated as operators:

Position operators: X, Y, Z

Momentum operators: P_x, P_y, P_z

These can be expressed as differential operators:

P_x = -ihbard/dx, P_y = -ihbard/dy, P_z = -ihbard/dz .

Components of Angular Momentum Operator

The angular momentum operator components are defined similarly to classical mechanics but involve the aforementioned position and momentum operators:

For example, the X component is given by:

$$ L_x = -ihbar(Yd/dz - Zd/dy) $$.

Importance of Commutation Relations

Commutation relations dictate the order of operations for quantum mechanical operators. If two operators commute (e.g., their commutation relation equals zero), their order does not matter; otherwise, it does.

Position operators commute with each other and so do momentum operators. However, position and momentum pairs do not commute (e.g., [X,P] ≠ 0) .

Finding Commutation Relations

To find the commutation relation between two angular momentum components like L_x and L_y :

$$ [L_x,L_y] = iL_z $$

This indicates that the order of operations matters significantly when calculating these relations .

Detailed Calculation Example

The calculation involves determining how different combinations of angular momentum components interact under commutation. For instance:

$$ [Y,P_Z] $$

shows that while some terms cancel out due to commuting properties, others do not because they involve non-commuting variables .

Understanding Commutation Relations in Quantum Mechanics

Commutation of Position and Momentum Operators

The commutation relation between position Z and momentum p_z is highlighted, showing that they do not commute:

[ [Z, p_z] = -ihbar ]

This leads to the expression involving cyclic order:

ihbar x p_y - y p_x [].

Angular Momentum Operators

The discussion extends to angular momentum operators L_x, L_y, L_z :

Their commutation relations are established:

[L_x, L_y] = ihbar L_z

[L_y, L_z] = ihbar L_x

Emphasis on maintaining cyclic order when calculating these relations [].

Uncertainty Principle Implications

The uncertainty principle for two operators A and B is introduced:

For angular momentum components:

It indicates that precise measurements of L_x and L_y cannot be made simultaneously.

Expressed as:

h^2/4 * E[L_z^2] , indicating incompatibility of observables [].

Commutation with Position Operators

Examination of the commutation between position operator X and angular momentum operators:

Results show that while some pairs commute (e.g., [X, P_X] = 0 ), others do not:

Non-zero relations include:

[P_X, L_Y] = ihbar P_Z; [P_X, L_Z] = -ihbar P_Y. [].

General Angular Momentum Relations

Transitioning to general angular momentum denoted by vector J:

Similar expressions for commutation relations are derived for components of J.

Notably,

J_s = J_x^2 + J_y^2 + J_z^2

which commutes with each component [].

Hermitian Properties and Eigenvalues

Discussion on the Hermitian nature of operators like J:

They possess real eigenvalues.

While Jx, Jy, and Jz can have simultaneous eigenfunctions with Js, they cannot share them among themselves due to non-commutativity [].

Raising and Lowering Operators Introduction

Introduction of raising ( J_+ ) and lowering ( J_- ) operators defined as follows:

J_+ = J_x + iJ_y

J_- = J_x − iJ_y

These facilitate calculations involving eigenstates [].

Commutation Relations Among New Operators

Establishment of new commutation relations using raising/lowering operators shows their utility in quantum mechanics.

[Js, J_+] = [Js, J_-] = 0

Commutation Relations and Eigenvalues in Quantum Mechanics

Commutation of Angular Momentum Operators

The commutation relation between J_z and J_+ is found to be hbar J_+ , while the relation for J_z and J_- yields -hbar J_- .

The expression for the product of operators J_+ J_- can be expressed in terms of J^2 and J_z , leading to a formulation involving non-commuting terms.

Expressions Involving Angular Momentum Operators

The relationship between angular momentum squared, represented as j^2 = j_x^2 + j_y^2 + j_z^2 , allows simplification of expressions involving these operators.

Using commutation relations, we derive eigenfunctions and eigenvalues for the operators acting on states denoted as alpha ( alpha ) and beta ( beta ).

Eigenstates and Eigenvalues Analysis

When applying the operator J_z , it modifies the state by changing beta, resulting in eigenvalues expressed as hbeta_pm 1 .

The action of raising ( J_+ ) or lowering ( J_- ) operators on states leads to new eigenstates with specific eigenvalues related to total angular momentum.

Constraints on Eigenvalue Relationships

It is established that there exists an upper limit for beta based on expectation values derived from angular momentum squared minus its z-component.

This leads to a conclusion that if we apply the raising operator multiple times, it will eventually reach a maximum state where further application results in zero.

Ladder Operator Dynamics

By analyzing how raising and lowering operators interact with states, we find relationships that define limits for alpha based on beta's maximum value.

A ladder analogy is used to describe transitions between states defined by their respective eigenvalues under operations performed by these angular momentum operators.

Final Formulation of Eigenstates

Two potential solutions arise from equations relating maximum and minimum values of beta; however, only one solution remains viable within physical constraints.

Ultimately, we establish that possible values for total angular momentum (J values ranging from integers to half-integers), along with corresponding M values spanning from negative to positive limits.

Summary of Results

The final results yield orthonormality conditions among eigenfunctions represented as delta functions.

Understanding the Operators in Quantum Mechanics

The Role of Raising and Lowering Operators

The discussion begins with the equation involving raising (J+) and lowering (J-) operators, leading to the identification of constants c_jm for both c_jm+ and c_jm- . It is noted that J+ is equivalent to J- , allowing for a reformulation of expressions.

The operators can be expressed in terms of angular momentum components, specifically J_s and J_z . This allows for calculations involving states represented by quantum numbers.

By manipulating these expressions, one can derive relationships between different states. When applying the raising operator on state |j,mrangle , it results in a new state while preserving the value of j .

Effects of Operators on Quantum States

The action of the raising operator ( J+ ) increases the magnetic quantum number ( m ), while the lowering operator ( J- ) decreases it. This distinction highlights their roles as operators that raise or lower states within a defined system.

Further exploration reveals how other angular momentum components like J_x and J_y can also be expressed using these raising and lowering operators. Notably, differences arise due to signs in their mathematical representations.

Expectation Values and Inner Products

A critical point is made regarding expectation values of operators such as J_y and J_x. These values are examined within simultaneous eigenstates of other operators, emphasizing how they alter states like |j,mrangle.