Time dependence of expectation values

TL;DR

The time dependence of the expectation values in quantum mechanics can be represented using the commutator of the operator and Hamiltonian, allowing for the understanding of conservation laws.

Transcript

PROFESSOR: It's a statement about the time dependence of the expectation values. It's a pretty fundamental theorem. So here it goes. You have d dt of the expectation value of Q. This is what we want to evaluate. We Now this would be d dt of integral psi star of x and t, Q psi of x and t. And the d dt acts on the two of them. So it gives you integra... Read More

Key Insights

• 🦾 The time dependence of expectation values in quantum mechanics is represented using the commutator of the operator and Hamiltonian.
• 👮 The commutator is essential for understanding the dynamics and conservation laws of observables.
• ⌛ Expectation values of quantities that commute with the Hamiltonian remain constant over time.

Questions & Answers

Q: How can the time dependence of the expectation values in quantum mechanics be represented?

The time dependence of the expectation values can be represented using the commutator of the operator and Hamiltonian, as shown in the Schrodinger equation. This allows for the understanding of how observables change over time.

Q: What is the significance of the commutator in quantum mechanics?

The commutator is crucial in quantum mechanics as it determines the dynamics and conservation laws of observables. It shows how the change in time of the expectation value is related to the commutator of the operator and Hamiltonian.

Q: How does the commutator relate to conservation laws in quantum mechanics?

Conservation laws in quantum mechanics are associated with quantities that commute with the Hamiltonian. If a quantity commutes with the Hamiltonian, its expectation value remains constant over time, resulting in conservation.

Q: Can you give an example of a conservation law in quantum mechanics?

A conservation law can be seen when a momentum operator commutes with the Hamiltonian. As a result, the expectation value of the momentum remains constant over time, leading to the conservation of momentum.

Summary & Key Takeaways

• The time dependence of the expectation value of Q can be evaluated using the Schrodinger equation and the commutator of Q and the Hamiltonian.

• The commutator plays a significant role in quantum mechanics and helps determine the dynamics and conservation laws of observables.

• If a quantity commutes with the Hamiltonian, its expectation value will remain constant over time.