Question about expectation value.

In summary: This theorem states that the expectation value of the time derivative of any operator is equal to the commutator of that operator with the Hamiltonian. In summary, the energy expectation value is always independent of time as long as the Hamiltonian is time-independent, which is a fundamental result in classical and quantum mechanics. This can be proven in general using the Ehrenfest theorem.
  • #1
cragar
2,552
3
It seems that the energy expectation value is independent of time.
I did it for an infinite square well. And when you time evolve your wave function
the time evolution cancels when you complex conjugate it and then do the integral.
<E>=<ψ|E|ψ> it seem like this might always independent of time, What do you guys think.
I guess I could try to prove it in general.
 
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  • #2
cragar said:
It seems that the energy expectation value is independent of time.

That's called energy conservation and should not be so surprising. It's true as long as the hamiltonian is time independent. And it generalized to the expectation value of any operator that commutes with the hamiltonian.
 
  • #3
If you have a time-independent Hamiltonian, energy is always conserved in your system. This is a very fundamental result and it works both in classical and quantum mechanics.
 
  • #4
cragar said:
It seems that the energy expectation value is independent of time.
I did it for an infinite square well. And when you time evolve your wave function
the time evolution cancels when you complex conjugate it and then do the integral.
<E>=<ψ|E|ψ> it seem like this might always independent of time, What do you guys think.
I guess I could try to prove it in general.

A more general result is the Ehrenfest theorem: http://en.wikipedia.org/wiki/Ehrenfest_theorem
 

FAQ: Question about expectation value.

What is an expectation value?

An expectation value is a calculation used in statistics and quantum mechanics to determine the average outcome of a particular event or measurement. It takes into account all possible outcomes and their associated probabilities.

How is the expectation value calculated?

The expectation value is calculated by multiplying each possible outcome by its associated probability and then summing up all of these products. This gives the average or most likely outcome of the event or measurement.

What is the significance of the expectation value?

The expectation value is significant because it provides a single value that represents the most likely outcome of a system. It allows scientists to make predictions about the behavior of a system and compare it to experimental results.

What is the difference between expectation value and average value?

The expectation value and average value are often used interchangeably, but there is a slight difference. The expectation value takes into account all possible outcomes and their associated probabilities, while the average value is simply the arithmetic mean of a set of values.

How is the expectation value used in quantum mechanics?

In quantum mechanics, the expectation value is used to predict the most likely outcome of a measurement on a quantum system. It is also used to calculate the probability of obtaining a particular measurement result.

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