Angular Momentum Mix (Griffiths)

A system is in the state

$|\psi>=(2/\sqrt{11})|11>+(1/\sqrt{17})|10>+(2i\sqrt{17})|2,-1>+$

$+(2/\sqrt{17})|31>+(2i/\sqrt{17})|32>$

What are the possible measurements for $L^2$, and $L_z$ and their probabilities? What is $<L_z>$?

Let $|∝β>$ be the common eigen vectors of operators $L^2$ and $L_z$. A state of the particle denoted by the wave function ψ can be written as a linear combination of the common eigen vectors

$|ψ>=∑_n A_n*|α_n β_n>$

For a particular common eigen vector $|αβ>$ one has the definition relations:

$L^2*|αβ> =α*|αβ>and L_z*|αβ> =β*|αβ>$

Since for an eigenfunction ψ one has

$L^2 |ψ>=l(l+1) \hbar^2*ψ$ and $L_z |ψ>=m \hbar*|ψ>$

It follows that $α=l(l+1)$ and $β=m$ if we make the convention $\hbar=1$

For the particular state given in text:

$|ψ> =2/√17*|11>+ 1/√17*|10>+2i/√17*|2,-1>+2/√17*|31>+2i/√17*|32>$

The possible measurements of the square of the angular momentum $L^2$ are in order:

$\hbar^2,\hbar^2,2\hbar^2,3\hbar^2,3\hbar^2$

The probabilities for each measurement are simply $P=∑_i|A_i |^2$ since the eigen vectors $|αβ>$ are orthogonal to each other.

The probabilities for each value are

-for $\hbar^2$ probability is $P_1=4/17+1/17=5/17$

-for $2\hbar^2$ the probability is $P_2=4/17$

-for $3hbar^2$ the probability is $P_3=4/17+4/17=8/17$

All probabilities add to 1 as expected.

The possible measurements of $L_z$ are in order

$\hbar,0,-\hbar,\hbar,2\hbar$

The probabilities for each value are

-for $\hbar$ the probability is $P_1=4/17+4/17=8/17$

-for 0 the probability is $P_0=1/17$

-for $-\hbar$ the probability is $P_(-1)=4/17$

-for $2\hbar$ the probability is $P_2=4/17$

The probabilities do not add to 1 since the state $-2\hbar$ is not present in the value of $ψ$.

The expectation value of $L_z$ is simply

$<L_z> =∑_n |A_n |^2*β=[4/17*1+1/17*0+4/17*(-1)+4/17*1+4/17*2] \hbar=$

$=12/17*\hbar$