Showing the expectation values of a system are real quantities

In summary, the homework statement is that a one-dimensional system is in a state described by the normalisable wave function Ψ(x,t). The expectation value of the position ⟨x⟩ is a real quantity, and the expectation value of the momentum in the x-direction ⟨p⟩ is a real quantity, too.
  • #1

Homework Statement



A one-dimension system is in a state described by the normalisable wave function Ψ(x,t) i.e. Ψ → 0 for x → ±∞.

(a) Show that the expectation value of the position ⟨x⟩ is a real quantity. [1]

(b) Show that the expectation value of the momentum in the x-direction ⟨p⟩ is a real quantity, too. Hint: using integration by parts and normalisability show that ⟨pˆx⟩ = ⟨p⟩∗. [4]


Homework Equations



1=N∫ψ*ψ.dx

<x>=∫ψ* xψ.dx over all space

<p>=∫ψ* -ih dψ/dx.dx over all space

The Attempt at a Solution



The difficult aspect of this for me is determining what the correct wave function is. Using the information given I assumed that the correct wave function was e-ax2/2 eiEt/h (where a/2 is an arbitrary constant) as it fits the above requirements (I could be wrong.)

However, upon normalising and calculating <x> and <p>, the values obtained will of course will be 0 and therefore real as my assumption was a symmetric wave function. This is all well and good, however the question explicitly states to use integration by parts to solve for <p>.

<p>=N∫(e-ax2/2 eiEt/h) -ih d/dx(e-ax2/2 e-iEt/h).dx over all space

which gave:

<p>=N iha∫xe-ax2.dx over all space

which cannot be integrated by parts as far as I understand-perhaps it can?. Have I got the wrong end of the stick somewhere in my thinking?

Thanks in advance.
 
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  • #2
You wave function is incorrect. Do not plug it in until you have done the math. You do not even need a "test" wavefunction for part a.
 
  • #3
Thanks for the prompt reply, but I don't quite follow. Could you clarify what you mean by 'not needing a wave function'? I feel I am approaching the problem from the wrong angle.
 
  • #4
Hmm, perhaps forget what I mentioned previously. You simply need to reconsider your wave function. Perhaps try something of the form [itex] \psi = e^{\pm i(kx-E t/ \hbar)} [/itex]. . .
 
  • #5




Thank you for your question. It seems like you have the right idea, but there are a few things that need to be clarified. First, the wave function given in the question is not necessarily the correct one, it is just an example of a normalizable wave function that satisfies the given conditions. So, you can use any other normalizable wave function as well.

Now, for part (a), to show that the expectation value of position is a real quantity, you can simply use the definition of the expectation value and the fact that the wave function is normalized. So, you should get:

<x>=∫Ψ* x Ψ dx = ∫Ψ* Ψ x dx = ∫|Ψ|2 x dx

Since the wave function is normalized, the integral of |Ψ|2 over all space is equal to 1. Therefore, <x> is a real quantity.

For part (b), you are on the right track. To show that the expectation value of momentum is a real quantity, you can use integration by parts as follows:

<p>=∫Ψ* (-ih dΨ/dx) dx = ih∫(dΨ*/dx) Ψ dx = ih[Ψ* Ψ] - ih∫Ψ* (dΨ/dx) dx

Since the wave function goes to 0 at infinity, the first term in the above equation is equal to 0. And by the same logic as in part (a), the integral of |Ψ|2 over all space is equal to 1. Therefore, <p> is a real quantity.

I hope this helps clarify your understanding. Keep up the good work!
 

1. What does it mean for an expectation value to be a real quantity?

An expectation value being a real quantity means that it is a numerical value that can be measured in the physical world, and it is not imaginary or complex in nature. This means that the expectation value is a tangible and observable quantity.

2. How is the reality of an expectation value of a system determined?

The reality of an expectation value of a system is determined by taking the average of all possible measurements of the system and comparing it to the theoretical prediction. If the two values match, it confirms the reality of the expectation value.

3. Can the expectation value of a system be a complex or imaginary number?

No, the expectation value of a system cannot be a complex or imaginary number because it represents a physical quantity that is measured in the real world. Complex or imaginary numbers do not have a physical manifestation.

4. Why is it important to show that the expectation values of a system are real?

It is important to show that the expectation values of a system are real because it confirms the validity of the theoretical model used to describe the system. It also allows for accurate predictions to be made about the behavior of the system and its physical properties.

5. Are there any exceptions where the expectation values of a system may not be real?

There are some cases where the expectation values of a system may not be real, such as in quantum mechanics where the uncertainty principle states that certain variables cannot have precise values. In these cases, the expectation values may be complex or imaginary, but they still hold physical significance in the theoretical model.

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