1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Hard Quantum Question

  1. Feb 21, 2008 #1
    1. The problem statement, all variables and given/known data

    Prove that


    <xp>=∫ψ* x(ħ/i)(∂/∂x) ψ dx

    nor <xp>=∫ψ* x(ħ/i)(∂/∂x)xψ dx

    is acceptable because both lead to imaginary value.Show that

    <xp>=∫ψ* x(ħ/i)(∂/∂x) ψ dx + ∫ψ* x(ħ/i)(∂/∂x)xψ dx leads to real value.Does

    <xp>=<x><p> ?

    2. Relevant equations
    3. The attempt at a solution

    Taking * of proposed <xp>=∫ψ* x(ħ/i)(∂/∂x) ψ dx and carrying out by parts integral,I get

    <xp>*= -iħ +<xp> ≠ <xp> .Hence, given expression for <xp> is not real where it should be so.

    [What I am woried about,even <xp> comes out to be imaginary, <xp>-<px>=iħ is still OK.Because <xp>*=<px>]

    Also,my attempts to show the second proposed expression leads to imaginary value failed:

    <xp>=∫ψ* x(ħ/i)(∂/∂x)xψ dx

    => <xp>*=∫ψ x(-ħ/i) (∂/∂x)xψ* dx

    =><xp>*=(iħ) ∫ψx (∂/∂x)xψ* dx =(iħ) [-∫xψ*(ψ+ x(∂ψ/∂x)) dx + 0]

    where I assumed [x²ψ*ψ] gives zero in both +∞ and -∞ limit.

    [I am not sure at this point too.As the term involves a factor x²]

    Can anyone suggest if I am going through the correct way?If I am doing wrong in integration, please show it.
  2. jcsd
  3. Feb 21, 2008 #2
    Given proposed expressions for [tex]\langle\ x \ p_x \rangle=\int^\infty_{\ - \infty}\psi\ast\ x \frac{\hbar}{i}[/tex][tex]\frac{\partial\psi}{\partial\ x }[/tex][tex]\ dx[/tex]

    Another given expression for [tex]\langle\ x \ p_x \rangle=\int^\infty_{\ - \infty}\psi\ast\ x \frac{\hbar}{i}[/tex][tex]\frac{\partial(\ x \psi)}{\partial\ x }[/tex][tex]\ dx[/tex]

    We are to show that neither is correct but the sum of the integrations is the correct expression for <xp_x>

    I started with taking * of [tex]\langle\ x \ p_x \rangle[/tex] of the first expression:

    [tex]\langle\ x \ p_x \rangle\ast=\frac{-\hbar}{i}\int_{\ - \infty}^\infty\psi\ x \frac{\partial\psi\ast}{\partial\ x }[/tex][tex]\ dx=[/tex][tex]\ i \hbar\ [\ - \int_{\ - \infty}^{\infty}\frac{\partial(\psi\ x )}{\partial\ x }\psi\ast\ dx\ + \(\psi\ast\ x \psi)_{\ - \infty}^{\infty}][/tex]

    Last term=0

    [tex]\langle\ x \ p_x \rangle\ast=\ i \hbar\[\ - \int_{\ - \infty}^\infty\psi\ast\ (\psi + \ x \frac{\partial\psi}{\partial\ x})\ dx \]=\ - \ i \hbar\ + \langle\ x \ p_x \rangle[/tex]

    But this is not [tex]\langle\ x \ p_x \rangle[/tex].So, we conclude given expression of [tex]\langle\ x \ p_x \rangle[/tex] is not correct because the expectation value must be a real quantity Q such that Q*=Q

    Note an interesting thing! Even if the expression is not correct,it correctly leads to <xp>-<px>=iħ as <xp>*=<px>

    But I am in trouble with the second integration.It appears that the conjugate of the second integral should be equal to iħ so that the sum of the two integrals be <xp>

    Can anyone please suggest anything?
    Last edited: Feb 22, 2008
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Hard Quantum Question
  1. Hard thermo question. (Replies: 3)