- #1

neelakash

- 511

- 1

## Homework Statement

Prove that

Neither

<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> ?

## Homework Equations

## 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.