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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
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