Question

consider a simple harmonic oscilator and its normalised eigenstates

Problem 5. Consider operators B and C, whose matrix representations in some basis are: ſo 10] [1ool 1 0 1 C=o oo Too-1 0 1 0 (a) (10 points) What are the possible values one can obtain if operator C is measured? (b) (10 points) What are the possible states after the measurement? (e) (10) points) Take the state in which C=1. In this state, what are (B) and (BP)? (d) (10 points) If the system is in a stated) and operator C is measured, what are the possible outcomes and their probabilities? (d) = 1 i

Answer 1

Solution:

(a)

Since $\small C$ is diagonal, we are in the $\small C$ basis and the diagonal elements are the $\small C$ eigenvalues, we have  $\small C \pm 1,0$.

(b)

The corresponding eigenvectors are:

C =1>= 10 = 11 > 0

0 1 -1

C = 0 >= 1 0 = 10 >

(c)

$\small \left \langle B \right \rangle=<1|B|1>=\left ( 1,0,0 \right )\frac{1}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0 \end{pmatrix}\begin{pmatrix} 1\\ 0\\ 0\\ \end{pmatrix}=0$

$\small \left \langle B^{2} \right \rangle=<1|B^{2}|1>=\left ( 1,0,0 \right )\frac{1}{2}\begin{pmatrix} 1 & 0 & 1\\ 0 & 2 & 0\\ 1 & 0 & 1 \end{pmatrix}\begin{pmatrix} 1\\ 0\\ 0\\ \end{pmatrix}=\frac{1}{2}$

(d)

We have

$\small |d>=\frac{1}{\sqrt{2}}|C=1>-\frac{i}{\sqrt{2}}|C=0>+0|C=-1>$

$\small P(C=1|d)=|<C=1|d>|^{2}=\frac{1}{2}$

$\small P(C=0|d)=|<C=0|d>|^{2}=\frac{1}{2}$

$\small P(C=-1|d)=|<C=-1|d>|^{2}=0$