Quantum mechanics - operator problem

[Brian-san]

Homework Statement

A system is characterized by two commuting operators $$\Omega_1,\Omega_2$$, each of whose eigenvalues are ±1. Thus the eigenkets are $$\left|\omega_1,\omega_2\right\rangle$$, where $$\omega_i=\pm 1$$. The system is passed through device D which measures $$\omega_1+\omega_2$$. It leaves intact those systems for which $$\omega_1+\omega_2\geq0$$, but rejects $$\omega_1+\omega_2<0$$.

a) Using the above notation for bras and kets, write the expression for operator D.
b) A second device D' has the property that
$$D'\left|\omega_1,\omega_2\right\rangle=\frac{1}{\sqrt{2}}\left[\left|\omega_1,-\omega_2\right\rangle+\left|-\omega_1,\omega_2\right\rangle\right]$$.
Give the expression for the operator D'. Show that it is Hermitian.
c) Determine the characteristic equation for D'.
d) What are it's eigenvalues and their multiplicities? Find corresponding eigenkets.
e) Find the expectation value of D' for a state obtained by the action of D on a state shich is an equal admixture of the four basis kets $$\left|\omega_1,\omega_2\right\rangle$$.

Homework Equations

An operator is Hermitian if it is it's own adjoint/is represented by a Hermitian matrix
The eigenvalues of an operator are the roots of it's characteristic polynomial.
The characteristic equation can be found by taking successive powers of the operator and finding a combination that produces zero.

The Attempt at a Solution

This is the one problem on the assignment I've just been staring at, without any idea of how to proceed. I've never seen a problem like this, either in my notes, textbook, or remember on from lectures. Probably the most bothersome part is not fully understanding the notation used. I assume that D and D' can be expressed in terms of the two operatos $$\Omega_1,\Omega_2$$, but can't figure out how to combine them, or what they mean on kets of the form $$\left|\omega_1,\omega_2\right\rangle$$. Once I get that, parts b, c and d should easily follow, as I know how to go about finding characteristic equations/eigenvalues/etc.

Lastly, I do not understand exactly what part e is asking. I'm familiar with calculating expectation values, but that was more of the usual things like position/momentum/energy operators. Help is greatly appreciated with this one.

 

[gabbagabbahey]

You don't really need to worry about expressing $D$ or $D'$ in terms of $\Omega_1$ and $\Omega_2$...the fact that the system is completely characterized by $\Omega_1$ and $\Omega_2$ with eigenvalues $\pm 1$, just tells you that any pure state in the system can be written as $|\psi\rangle=|\omega_1,\omega_2\rangle$, where $\omega_1$ and $\omega_2$ (the eigenvalues of $\Omega_1$ and $\Omega_2$, respectively) each will be $\pm1$ (depending on which pure state the system is in)...

What part (a) asks you to do, is find an equation that describes $D$, the same way that the equation given in part (b) describes $D'$...If an operator measures the quantity $\omega_1+\omega_2$, what is the definition of its projection operator $\mathbb{P}_{\omega_1+\omega_2}$?...What, in terms of $\mathbb{P}_{\omega_1+\omega_2}$, will happen to the state $|\psi\rangle=|\omega_1,\omega_2\rangle$ when $D$ measures $\omega_1+\omega_2$?...What does the problem statement tell you that result is?

(Remember, any mixed state can be written as a superposition of the pure states $|\omega_1,\omega_2\rangle$, so determining the action of $D$ on the pure states, completely determines its action on the system)

 

[diazona]

Actually, I think the problem is asking for the expressions of $D$ and $D'$ in terms of $\Omega_1$ and $\Omega_2$. The relation given in part (b) is just a constraint, not an expression that defines the operator... and if by "expression for the operator" they did mean something of that form, then what would be the point of asking for an expression for $D'$ in part (b)?

For part (a), I would start by considering the condition given in the problem, that device/operator D measures the quantity $\omega_1 + \omega_2$. Mathematically, that means (if I'm not too tired to think this through properly) that the eigenvalues of $D$ are $\omega_1 + \omega_2$. Hopefully you can guess an expression for $D$ that satisfies that condition.

 

[gabbagabbahey]

Actually, I think the problem is asking for the expressions of $D$ and $D'$ in terms of $\Omega_1$ and $\Omega_2$. The relation given in part (b) is just a constraint, not an expression that defines the operator... and if by "expression for the operator" they did mean something of that form, then what would be the point of asking for an expression for $D'$ in part (b)?

I'm fairly sure that the expressions asked for will be expected in the form

$$D=\sum_{i,j,k,l=\pm1}\alpha_{ijkl}|i,j\rangle\langle k,l|$$

and

$$D'=\sum_{i,j,k,l=\pm1}\alpha'_{ijkl}|i,j\rangle\langle k,l|$$

(Basically, the expansions of the operators in the eigenbasis of $\Omega_1$ and $\Omega_2$)


The constants $\alpha_{ijkl}$ can be determined in a straightforward manner once the expansion of $D|\omega_1,\omega_2\rangle$ is found...I don't see how either $D$ or $D'$ could be written in terms of $\Omega_1$ and $\Omega_2$.

For example, if I found

$$G|\omega_1,\omega_2\rangle=\left\{\begin{array}{lr}(\omega_1+2\omega_2)|-\omega_2,\omega_1\rangle &, \omega_1=\omega_2 \\0 &, \omega_1\neq\omega_2\end{array}$$

I would calculate the following,

$$G|1,1\rangle=3|-1,1\rangle$$
$$G|1,-1\rangle=0$$
$$G|-1,1\rangle=0$$
$$G|-1,-1\rangle=-|1,-1\rangle$$


And hence, I would conclude $G=3|-1,1\rangle\langle 1,1|-|1,-1\rangle\langle -1,-1|$