Monday, May 13, 2024

Solutions to Quantum Mechanics Problems

 Solutions to the state function problems from last week:

1) Given eigenvalues a and eigenvectors un of the operator  A^   find the eigenvectors if:

A^  is of the diagonal form (e.g. for the matrix)

 (a1 …….0)

 (0……  a2)

Solution:

| u1 >   

u2 >   


2)  In a quantum mechanics exam a student writes for the normalization condition:y | y > =    ò |y | x >| dx =       1

Rewrite the above equation as it should be given

Solution:

< y | y > =   ò y | x ><x |y > dx =     ò |y | x >|2  dx =    1

3) Prove that the eigenvalue a  for an observable operator A^  is real.

Solution:

If a is any eigenvalue of  A^ then:

A^ a >  =    aa >

A^ (x ,   x) <xa >   =    a<xa >

Then we can multiply  by <a | x > and integrate:

ò <a | x > A^ (x ,   x) <x |a > dx =  a ò <a | x ><x |a > dx

Note the left hand side is real and <a | a >  is the normalization integral, which is real by def.  Hence, a is real.

4) Let a1  and  a2  be different eigenvalues of  A^ .   Find: 

A^ a1 >  and A^ a2 >  and thence show:

<a1| A^ |a2 >    =   a<a1 | a2  >

Solution:

A^ a1 >  =       a1 a1 >

A^ a2 >  =       a2 a2 >

Then:

<a2| A^ |a1 > =  a<a2 | a1  >

<a1| A^ |a2 >    =  a<a1 | a2  >

Taking the complex conjugate and using:

< y |A^ |y > =   < y |A^ |y >*

So:

<a1| A^ |a2 >    =   a<a1 | a2  >


5) Show that u(x) = exp -(½2)   is an eigenfunction of the operator:

A^ (x ,   x)  =  (  2 /  x2   -  x)


Solution:

A^ (x ,   xn(x)    =   an  n(x)

So:

(  2 /  x2   -  x) exp -(½2)   =   a  exp -(½2)   

2   exp -(½2) -   exp -(½2) -  2   exp -(½2)  =  exp -(½2)  

-   exp -(½2)   =   exp -(½2)  

Then:  a = -1


No comments: