Quantum mechanics: Expectation values

In summary, when one wants to find the expectation value for a function at a point in space, they use the integral. However, they must set the time to 0 when doing this, or else the time dependence will become apparent.
  • #1
Niles
1,866
0

Homework Statement


Hi all.

Let's say that i have a wave function
[tex]\Psi (x,t) = A \cdot \exp ( - \lambda \cdot \left| x \right|) \cdot \exp ( - i\omega t)[/tex]

I want to find the expectation value for x. For this I use
[tex]\left\langle x \right\rangle = \int_{ - \infty }^\infty x \left| \Psi \right|^2 dx[/tex].

Why must I set t = 0 when doing this?
 
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  • #2
Niles said:

Homework Statement


Hi all.

Let's say that i have a wave function
[tex]\Psi (x,t) = A \cdot \exp ( - \lambda \cdot \left| x \right|) \cdot \exp ( - i\omega t)[/tex]

I want to find the expectation value for x. For this I use
[tex]\left\langle x \right\rangle = \int_{ - \infty }^\infty x \left| \Psi \right|^2 dx[/tex].

Why must I set t = 0 when doing this?

You don't.

Just leave t arbitrary and you will see that it will drop out by itself.
 
  • #3
Yes, you are right. But that is when I have normalized the wave function first.

In my book, when he normalizes this function he uses t=0, so A = sqrt(lambda). This way he uses t=0 when finding <x>.

But I can't do both (both use t=t and t=0). So what is more correct?
 
  • #4
Niles said:
Yes, you are right. But that is when I have normalized the wave function first.

In my book, when he normalizes this function he uses t=0, so A = sqrt(lambda). This way he uses t=0 when finding <x>.

But I can't do both (both use t=t and t=0). So what is more correct?

? The normalization is independent of t. exp(i*w*t) times it's complex conjugate is 1. For all t.
 
  • #5
Dick said:
? The normalization is independent of t. exp(i*w*t) times it's complex conjugate is 1. For all t.

My mistake. I just squared the whole expression, when I should have multiplied with the conjugate (we are dealing with complex numbers, of course).

Thanks to both of you.
 
  • #6
Niles said:
My mistake. I just squared the whole expression, when I should have multiplied with the conjugate (we are dealing with complex numbers, of course).

Thanks to both of you.

You are welcome.

By the way, it is a general property of wavefunctions: if you normalize them at any time, they will remain normalized at all times (this assumes that the wavefunction is square integrable, that the potential is real, etc). So one can in principle fix the time to be some value (any value!) before normalizaing but this is never necessary because the time dependence *always* drops out when we normalize.
 

1. What is an expectation value in quantum mechanics?

An expectation value in quantum mechanics is the average value that a physical quantity will have when measured in a quantum system. It is calculated by taking the weighted average of all possible measurement outcomes, using the probabilities determined by the system's wave function.

2. How is an expectation value calculated?

An expectation value is calculated by taking the integral of the observable (physical quantity) multiplied by the probability density function, which is determined by the system's wave function. This integral represents the weighted average of all possible measurement outcomes.

3. What does the expectation value tell us about a quantum system?

The expectation value provides information about the most likely outcome of a measurement in a quantum system. It can also give us insights into the distribution of possible measurement outcomes and the overall behavior of the system.

4. Can the expectation value of a physical quantity change over time?

Yes, the expectation value of a physical quantity can change over time in a quantum system. This is because the wave function, and therefore the probability density function used to calculate the expectation value, can change over time due to the system's dynamics.

5. How is the concept of expectation value used in quantum mechanics?

The concept of expectation value is used to make predictions about the behavior of quantum systems. It allows us to calculate the most probable outcome of a measurement and understand the overall behavior of the system. Expectation values are also important in the calculation of other key concepts in quantum mechanics, such as uncertainty and superposition.

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