- #1
s_gunn
- 34
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normalize the wave function and more! Please help!
i) Normalize the wave function
ii) Calculate <x>
iii) Calculate [tex]<x^{2}>[/tex]
iv) What would happen if a < 0?
[tex]\psi\left(x\right) = N\left(1+i\right)exp\left(-a|x|\right)[/tex], for -inf < x < inf and a > 0
It would take ages for me to work out the latex for all my steps (I'm new to latex!) so I'll do what i need to and hope someone can help!
First:
[tex]^{inf}_{-inf}\int 2N^{2}e^{-2a|x|}dx[/tex]
[tex]=N^{2}\left[-e^{-2a|x|}\right]^{inf}_{-inf}[/tex]
so: [tex]\frac{-N^{2}}{a}=1[/tex]
so: [tex]N=\sqrt{\frac{-1}{a}}[/tex]
therefore:
[tex]\psi\left(x\right) = \sqrt{\frac{-1}{a}}\left(1+i\right)exp\left(-a|x|\right)[/tex]
ii+iii) for the expectation values, I got both equalling zero
iv) and if a < 0, you'd get exponential growth as x approaches infinity (+ and -)
Is this right??!
I get so confused when the limits are infinity!
Homework Statement
i) Normalize the wave function
ii) Calculate <x>
iii) Calculate [tex]<x^{2}>[/tex]
iv) What would happen if a < 0?
Homework Equations
[tex]\psi\left(x\right) = N\left(1+i\right)exp\left(-a|x|\right)[/tex], for -inf < x < inf and a > 0
The Attempt at a Solution
It would take ages for me to work out the latex for all my steps (I'm new to latex!) so I'll do what i need to and hope someone can help!
First:
[tex]^{inf}_{-inf}\int 2N^{2}e^{-2a|x|}dx[/tex]
[tex]=N^{2}\left[-e^{-2a|x|}\right]^{inf}_{-inf}[/tex]
so: [tex]\frac{-N^{2}}{a}=1[/tex]
so: [tex]N=\sqrt{\frac{-1}{a}}[/tex]
therefore:
[tex]\psi\left(x\right) = \sqrt{\frac{-1}{a}}\left(1+i\right)exp\left(-a|x|\right)[/tex]
ii+iii) for the expectation values, I got both equalling zero
iv) and if a < 0, you'd get exponential growth as x approaches infinity (+ and -)
Is this right??!
I get so confused when the limits are infinity!