91Ó°ÊÓ

What would it mean for an observable \(Q\) to be conserved, in quantum mechanics? At a minimum, the expectation value of \(Q\) should be constant in time, for any state \(\Psi\). The criterion for this (assuming \(Q\) has no explicit time dependence) is that \(\hat{Q}\) commute with the Hamiltonian (Equation 3.148). But we'd like something more: The probability \(\left|c_{n}\right|^{2}\) of getting any particular eigenvalue \(\left(\lambda_{n}\right)\) of \(\hat{Q}\) should be independent of \(t\). Show that this, too, is guaranteed by the condition \([\hat{H}, \hat{Q}]=0\). (Assume that the potential energy is independent of \(t\), but do not assume \(\Psi\) is a stationary state.) Hint: \(\hat{Q}\) and \(\hat{H}\) are compatible observables, so they have a complete set of simultaneous eigenvalues.

Step by step solution

01

Understanding the Problem

We are required to show that if an observable \( \hat{Q} \) is conserved in a quantum system, then the probability \( \left| c_{n} \right|^2 \) of measuring any particular eigenvalue \( \lambda_n \) of \( \hat{Q} \) remains constant over time. Specifically, we must demonstrate that \([\hat{H}, \hat{Q}] = 0\) ensures this property.

02

Analyzing the Commutation Relation

If the operators \( \hat{H} \) (Hamiltonian) and \( \hat{Q} \) commute, \([\hat{H}, \hat{Q}] = 0\), it means that \( \hat{H} \) and \( \hat{Q} \) can have a simultaneous set of eigenstates. This implies that these operators are compatible, and each eigenstate of \( \hat{Q} \) is an eigenstate of \( \hat{H} \), and vice versa.

03

Examining the Time Evolution

In quantum mechanics, the time evolution of a state \( |\Psi(t)\rangle \) is governed by the Schrödinger equation: \( i\hbar \frac{d}{dt}|\Psi(t)\rangle = \hat{H}|\Psi(t)\rangle \). If \( \hat{Q} \) and \( \hat{H} \) commute, the eigenstates of \( \hat{Q} \) evolve without mixing among different eigenvalues of \( \hat{Q} \), due to their shared eigenbasis with \( \hat{H} \).

04

Demonstrating Constant Probabilities

Given that each eigenstate of \( \hat{Q} \) is an eigenstate of \( \hat{H} \), if \( |\Psi(0)\rangle \) is a linear combination of energy eigenstates, its form remains invariant with respect to \( \hat{Q} \) over time. Therefore, the probability \( \left| c_{n}(t) \right|^2 \) of finding the system in the state corresponding to \( \lambda_n \) stays constant, as these probabilities do not change due to no 'mixing' caused by \( \hat{H} \).

05

Conclusion

Thus, because both \( \hat{H} \) and \( \hat{Q} \) share a complete set of simultaneous eigenstates, the commutation relation \([\hat{H}, \hat{Q}] = 0\) guarantees that the probability associated with each eigenvalue of \( \hat{Q} \) remains constant over time.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Observable

In quantum mechanics, an observable is essentially any physical quantity that can be measured, such as position, momentum, or energy. Each observable is associated with a particular operator, denoted typically by a hat symbol, for example, \( \hat{Q} \). Observables play a critical role in determining the characteristics of a quantum system.

Understanding observables is crucial because:

• They allow us to predict measurement outcomes.
• They are tied to operators that have corresponding eigenvalues and eigenstates.
• The measurement of an observable results in an eigenvalue of its operator, and this process leaves the system in the corresponding eigenstate.

The properties of observables are linked to their operators' mathematical structure, which includes aspects such as commutation relations.

Commutation Relation

A commutation relation in quantum mechanics indicates how two operators interact with each other through their products. If two operators \( \hat{A} \) and \( \hat{B} \) satisfy the commutation relation \( [\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A} = 0 \), these operators are said to commute.

This is significant for a few reasons:

• Commuting operators have a complete set of simultaneous eigenstates. This means measurements of one do not disturb the other.
• Commutation implies the possibility of measuring both observables with certainty and simultaneously.
• In the context of conservation laws, if an observable commutes with the Hamiltonian, it means the observable is conserved over time.

In our original exercise, \( \hat{Q} \) (as an observable) commuting with \( \hat{H} \) (the Hamiltonian) ensures that the eigenvalues of \( \hat{Q} \) do not change over time, reflecting a deeper consistency in the quantum system.

Time Evolution

Time evolution in quantum mechanics describes how a quantum state changes over time. It is controlled by the Schrödinger equation, \( i\hbar \frac{d}{dt}|\Psi(t)\rangle = \hat{H}|\Psi(t)\rangle \), where \( \hat{H} \) is the Hamiltonian, the energy operator of the system.

The key aspects of time evolution include:

• The Hamiltonian dictates the rate and manner of the system's evolution.
• If a state is an eigenstate of the Hamiltonian, it evolves with a simple phase factor, meaning its probability content remains unchanged.
• For conserved observables, the commutation of their operators with the Hamiltonian ensures the state’s evolution does not change their probabilities over time.

Thus, if \( \hat{Q} \) and \( \hat{H} \) commute, the eigenstates of \( \hat{Q} \) will maintain their identity over time without mixing, which is key to maintaining constant probabilities for measurements.

Eigenvalue Problem

The eigenvalue problem is a core concept in understanding the behavior of quantum systems. It involves finding the eigenvalues and eigenstates of an operator, which represent possible measurement outcomes and the associated quantum states. The eigenvalue equation for an operator \( \hat{Q} \) is expressed as \( \hat{Q}|\phi\rangle = \lambda|\phi\rangle \).

Main points to understand about eigenvalue problems are:

• Eigenvalues \( \lambda \) of an operator correspond to possible results from measuring the associated observable.
• Eigenstates \( |\phi\rangle \) are the states in which the system will be found post-measurement.
• If \( [\hat{H}, \hat{Q}] = 0 \), the eigenstates of \( \hat{Q} \) do not change over time, ensuring the system remains unaltered with respect to this observable.

Therefore, the constancy of measurement probabilities over time, as discussed in our exercise, is guaranteed by solving the eigenvalue problem with the operator \( \hat{Q} \) that commutes with \( \hat{H} \). This ensures all potential measurement results, or eigenvalues, \( \lambda_n \), retain constant probability throughout the quantum process.