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thepopasmurf
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I'm trying to teach myself quantum mechanics using a book I got. I made an attempt at one of the questions but there are no solutions or worked examples so I'm wondering if I got it right.
Here it goes
Suppose an observable quantity corresponds to the operator [tex]\hat{B}= -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}[/tex].
For a particular system, the eigenstates of this operator are
[tex]\Psi(x)=Asin\frac{n\pi x}{L}[/tex], where n = 1,2,3,...; A is the normalisation constant
Determine the eigenvalues of [tex]\hat{B}[/tex] for this case
[tex]\hat{A}\psi_{j}=a_{j}\psi_{j}[/tex] I think
I used the operator on [tex]\psi[/tex] and differenciated twice to get
[tex]\frac{\hbar^2 n^2 \pi^2}{2mL^2}ASin\frac{n\pi x}{L}[/tex]
this corresponds to [tex]a_j\psi_j[/tex] so my answer for the eigenvalues is
[tex] \frac{\hbar^2 n^2 \pi^2}{2mL^2} [/tex]
This is my first attempt at anything like this so any help is welcome
Here it goes
Homework Statement
Suppose an observable quantity corresponds to the operator [tex]\hat{B}= -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}[/tex].
For a particular system, the eigenstates of this operator are
[tex]\Psi(x)=Asin\frac{n\pi x}{L}[/tex], where n = 1,2,3,...; A is the normalisation constant
Determine the eigenvalues of [tex]\hat{B}[/tex] for this case
Homework Equations
[tex]\hat{A}\psi_{j}=a_{j}\psi_{j}[/tex] I think
The Attempt at a Solution
I used the operator on [tex]\psi[/tex] and differenciated twice to get
[tex]\frac{\hbar^2 n^2 \pi^2}{2mL^2}ASin\frac{n\pi x}{L}[/tex]
this corresponds to [tex]a_j\psi_j[/tex] so my answer for the eigenvalues is
[tex] \frac{\hbar^2 n^2 \pi^2}{2mL^2} [/tex]
This is my first attempt at anything like this so any help is welcome