Solutions to the state function problems from last week:
1) Given eigenvalues an 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 > = a| a >
A^ (x , ¶ / ¶ x) <x| a > = a<x| a >
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 > = a2 <a1 | a2 >
Solution:
A^ | a1 > = a1 | a1 >
A^ | a2 > = a2 | a2 >
Taking the complex conjugate and using:
< y
|A^ |y > = < y |A^ |y >*
So:
<a1| A^ |a2 > = a2 <a1 | a2 >
5) Show that u(x) = exp -(½x 2) is an eigenfunction of the operator:
A^ (x , ¶ / ¶ x) = ( ¶ 2 / ¶ x2 - x2 )
Solution:
A^ (x , ¶ / ¶ x) u n(x) = an u n(x)
So:
( ¶ 2 / ¶ x2 - x2 ) exp -(½x 2) = a exp -(½x 2)
x 2 exp -(½x 2) - exp -(½x 2) - x 2 exp -(½x 2) = a exp -(½x 2)
- exp -(½x 2) = a exp -(½x 2)
Then: a = -1
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