Principles of Quantum Mechanics · Problem 4.2.1

The Postulates—a General Discussion — R. Shankar

Physics — Quantum Mechanics · quantum-operators-angular-momentum

Problem

Consider the following operators on a Hilbert space

\mathbb{V}^3(C)

L_x = \frac{1}{2^{1/2}}\begin{bmatrix}0 & 1 & 0\\1 & 0 & 1\\0 & 1 & 0\end{bmatrix}, \quad L_y = \frac{1}{2^{1/2}}\begin{bmatrix}0 & -i & 0\\i & 0 & -i\\0 & i & 0\end{bmatrix}, \quad L_z = \begin{bmatrix}1 & 0 & 0\\0 & 0 & 0\\0 & 0 & -1\end{bmatrix}

(1) What are the possible values one can obtain if

L_z

is measured? (2) Take the state in which

L_z=1

. In this state what are

\langle L_x\rangle

\langle L_x^2\rangle

, and

\Delta L_x

? (3) Find the normalized eigenstates and the eigenvalues of

L_x

in the

L_z

basis. (4) If the particle is in the state with

L_z=-1

, and

L_x

is measured, what are the possible outcomes and their probabilities? (5) Consider the state

|\psi\rangle = \begin{bmatrix}1/2\\1/2\\1/2^{1/2}\end{bmatrix}

in the

L_z

basis. If

L_z^2

is measured in this state and a result

+1

is obtained, what is the state after the measurement? How probable was this result? If

L_z

is measured, what are the outcomes and respective probabilities? (6) A particle is in a state for which the probabilities are

P(L_z=1)=1/4

P(L_z=0)=1/2

, and

P(L_z=-1)=1/4

. Convince yourself that the most general, normalized state with this property is

|\psi\rangle = \frac{e^{i\delta_1}}{2}|L_z=1\rangle + \frac{e^{i\delta_2}}{2^{1/2}}|L_z=0\rangle + \frac{e^{i\delta_3}}{2}|L_z=-1\rangle

It was stated earlier on that if

|\psi\rangle

is a normalized state then the state

e^{i\theta}|\psi\rangle

is a physically equivalent normalized state. Does this mean that the factors

e^{i\delta_i}

multiplying the

L_z

eigenstates are irrelevant? [Calculate for example

P(L_x=0)