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majeka
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I have been asked to "find a solution to Ehrenfest's Theorem" (in this case for the expectation value of position, of a particle confined to a circle). What does this mean - what kind of answer should i find?
Ehrenfest's Theorem is a mathematical theorem that relates the time evolution of a quantum mechanical expectation value to the classical equations of motion. It states that the rate of change of the expectation value of a quantum mechanical operator is equal to the expectation value of the commutator of the operator with the Hamiltonian.
The position expectation value is the average position of a particle in a quantum system, calculated by taking the expectation value of the position operator. It represents the most likely position of the particle when measured.
To solve Ehrenfest's Theorem, you first need to determine the Hamiltonian of the system, which represents the total energy of the system. Then, you calculate the commutator of the position operator with the Hamiltonian. Finally, you take the expectation value of this commutator to find the rate of change of the position expectation value.
Solving Ehrenfest's Theorem allows us to understand the relationship between classical mechanics and quantum mechanics. It shows that, on average, a quantum system will behave similarly to a classical system, but with some key differences due to the inherent randomness of quantum mechanics.
Yes, there are limitations to Ehrenfest's Theorem. It only applies to systems that are in a stable state, and it does not take into account any quantum effects such as superposition or entanglement. Additionally, it is an approximation and may not accurately predict the behavior of highly complex quantum systems.