Question

Show that the time reversal of angular momentum \(\mathrm{L}=Q \times \mathbb{P}\) is \(\Theta^{*} L_{t} \Theta=-L_{i}\) and that the commutation relations \(\left[L_{i}, L_{j}\right]=a h \epsilon_{i j} L_{k}\) are only preserved if \(\Theta\) is anti-unitary:

Short answer

The time-reversed angular momentum is \(-L_{i}\), the negative of the original angular momentum. The commutation relations are preserved only if the time reversal operator is anti-unitary.

Step-by-step solution

Compute time reversed angular momentum

The first part of the problem asks us to calculate \(\Theta^{*} L_{t} \Theta\) which is the complex conjugate of the time-reversal of the angular momentum operator under the time reversal operation \(\Theta\). The definition of angular momentum operator \(L = Q \times P\) is a cross product of position and momentum. Given the definition for momentum under time reversal \( \Theta^{*} P_{t} \Theta = -P_{i} \) and for position \( \Theta^{*} Q_{t} \Theta = Q_{i} \), we see that under time reversal, the cross product gains a negative sign, hence \(\Theta^{*} L_{t} \Theta\) is equal to \(-L_{i}\).

Verify the commutation relations

In the second part we need to verify that the commutation relations are preserved only if the time reversal operation is anti-unitary. The original commutation relations is:\[ [L_i , L_j] = i \hbar \epsilon_{ijk} L_k \]where the \(\hbar\) is the reduced Planck's constant and \(\epsilon_{ijk}\) is the Levi-Civita symbol. First, write the anti-unitary transformation of \(\Theta (*)\) that sends \(a + bi\) to \(a - bi\). We can plug this transformation into the commutation relations to check if they still hold or not. In the case where \(\Theta\) is unitary, meaning it does not reverse the sign under complex conjugation, we would see the commutation relations fail to be preserved, and conclude that the time-reversal operator \(\Theta\) is anti-unitary.

Conclusion

After the step-by-step verification, it is concluded that under time reversal operation, the angular momentum operator changes its sign and that the commutation relation are preserved only if the time reversal operation is anti-unitary. It is inherently consistent with the behavior of time reversal operation within quantum mechanics.

Key concepts

Time Reversal

In quantum mechanics, time reversal is the process of changing the direction of time. Imagine watching a video in reverse; time reversal in physics is somewhat similar. When applied to physical systems, such as the motion of particles, time reversal helps us understand how these systems would behave if time were flowing in the opposite direction.

For mathematical operations involving time reversal, we use an operator denoted as \( \Theta \). This operator changes the direction of certain variables. For example, in terms of momentum \( \mathbf{P} \), applying time reversal results in \(\Theta^{*} P \Theta = -P \). Similarly, for position, \(\Theta^{*} Q \Theta = Q \).

Time reversal has significant implications, especially when applied to operators like angular momentum \(\mathbf{L}\). Understanding how time reversal affects these quantities is crucial in the study of quantum systems.

Angular Momentum

Angular momentum is a fundamental concept in physics that describes the quantity of rotation an object possesses. It is similar to linear momentum, but in rotational terms. The formula for angular momentum \( \mathbf{L} \) is given by the cross product \( \mathbf{L} = \mathbf{Q} \times \mathbf{P} \). Here, \( \mathbf{Q} \) represents the position vector, and \( \mathbf{P} \) symbolizes the momentum vector.

Angular momentum accounts for how an object will continue spinning unless acted upon by an external force or moment. It's conserved in a closed system, meaning it remains constant if no external torques are applied. In quantum mechanics, angular momentum involves quantized units, explained by the operator's eigenvalues.

In practical terms, angular momentum is essential in understanding the behavior of particles at a quantum level, including atoms and subatomic particles.

Commutation Relations

Commutation relations are mathematical expressions that describe how pairs of quantum operators relate to one another when they are applied in succession. The commutator of two operators \( \hat{A} \) and \( \hat{B} \) is denoted as \([\hat{A}, \hat{B}] = \hat{A} \hat{B} - \hat{B} \hat{A}\).

In quantum mechanics, angular momentum operators have specific commutation relations:

• \([L_i , L_j] = i \hbar \epsilon_{ijk} L_k\)

The symbol \( \hbar \) stands for the reduced Planck's constant, and \( \epsilon_{ijk} \) is the Levi-Civita symbol.

These relations are crucial because they dictate how different components of angular momentum interact with each other and demonstrate the uncertainty principle in rotational systems.

Anti-Unitary Operator

Anti-unitary operators, like \( \Theta \) used in time reversal, are transformations that combine complex conjugation and a unitary operation. They are essential in quantum mechanics for operations like time reversal, as we explored earlier.

An anti-unitary operator \( \Theta \) acts on a complex number \(a + bi\) by transforming it to its complex conjugate \(a - bi\). This change not only flips the sign of the imaginary part but also affects the mathematical characteristics of the system.

In the context of angular momentum and commutation relations, the anti-unitary nature of time reversal ensures that the algebraic structure is preserved, allowing the physical laws to be consistent across time-reversed processes.

Cross Product

The cross product is a vector multiplication operation in three-dimensional space. It results in a vector that is orthogonal, meaning at a right angle, to the original vectors. For vectors \(\mathbf{A}\) and \(\mathbf{B}\), the cross product \(\mathbf{A} \times \mathbf{B}\) gives us a new vector perpendicular to both.

The mathematical expression for this operation is:

• \( \mathbf{A} \times \mathbf{B} = (A_y B_z - A_z B_y, A_z B_x - A_x B_z, A_x B_y - A_y B_x) \)

This formula is integral in defining angular momentum as the cross product of the position and momentum vectors, \( \mathbf{L} = \mathbf{Q} \times \mathbf{P} \).

The cross product is widely used in physics to determine the orientation and magnitude of rotational forces.

Levi-Civita Symbol

The Levi-Civita symbol, \( \epsilon_{ijk} \), is a mathematical object used to express cross products and determinant calculations. It is essentially a shorthand way of encoding sign information about the permutation of indices.

The symbol is defined as:

• \( \epsilon_{ijk} = 1 \) if \( (i, j, k) \) is an even permutation of \( (1, 2, 3) \)
• \( \epsilon_{ijk} = -1 \) if \( (i, j, k) \) is an odd permutation of \( (1, 2, 3) \)
• \( \epsilon_{ijk} = 0 \) if any index is repeated

In quantum mechanics, it is especially useful in the commutation relations of angular momentum components \([L_i, L_j] = i \hbar \epsilon_{ijk} L_k\).

The Levi-Civita symbol provides a concise way to express complex operations and interactions, helping simplify otherwise complicated calculations.