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## Homework Statement

Consider two pairs of operators X

_{α}, P

_{α}, with α=1,2, that satisfy the commutation relationships [X

_{α},P

_{β}]=ihδ

_{αβ},[X

_{α},X

_{β}]=0,[P

_{α},P

_{β}]=0. These are two copies of the canonical algebra of the phase space.

a) Define the operators $$a_\alpha = \frac{1}{\sqrt{2\hbar}}(X_\alpha+ip_\alpha)$$ and $$a_\alpha^\dagger=\frac{1}{sqrt{2\hbar}}(X_\alpha-iP_\alpha)$$.

Show that those satisfy the commutation relationships [a

_{α},a

_{β}

^{†}]=δ

_{αβ}. These are the creation and annihilation operators for two independent harmonic oscillators.

b) The number operators are N

_{α}=a

_{α}

^{†}a

_{α}, with a=1,2 (there is no implicit sum). Considere the states $$|n_1,n_2\rangle_H=\frac{(a_1^\dagger)^{n_1}}{\sqrt{n_1!}}\frac{(a_2^\dagger)^{n_2}}{\sqrt{n_2!}}|0,0\rangle_H$$, where |n

_{1},n

_{2}) is the state of excitement n

_{1}in the first oscillator and n

_{2}in the second oscillator, while |0,0) is the vacuum of both oscillators. Show that n

_{1}and n

_{2}are the eigenstates of N

_{1}, N

_{2}. What are their eigenvalues?

c) Define the operators $$J^a=\frac{\hbar}{2}\sum_{\alpha,\beta}a_\alpha^\dagger\sigma_{\alpha\beta}^aa_\beta$$ and $$J^0=\frac{\hbar}{2}\sum_\alpha a_\alpha^\dagger a_\alpha$$, with a=1,2,3 and where σ

_{α,β}

^{a}is the (α,β) entry of the Pauli matrix σ

^{a}. Show that the three matrixes J

^{a}satisfy de commutation relationships [J

^{a},J

^{b}]=ih∑

_{c}ε

^{abc}J

^{c}

d) Show that $$J^2=\sum_a (J^a)^2=J^0(J^0+1)$$

e) If the operators J

^{a}satisfy the algebra of angular momentum, a base of the space of states has to consist of states of the form |J,M)

_{S}, simultaneously eigenstates of J

^{2},J

_{3}, with M=-J,-J+1,...,J-1,J and possibly various values of J. Consider the states $$|J,M\rangle_S=|J+M,J-M\rangle_H=\frac{(a_1^\dagger)^{J+M}}{\sqrt{n_1!}}\frac{(a_2^\dagger)^{J-M}}{\sqrt{n_2!}}|0,0\rangle_H$$. These are the Schwinger states of angular momentum. That's why we use the sub indexes H and S to distinguish between the states of the harmonic oscillators and those of Schwinger. Show that these are eigenstates of J

^{2}and J

_{3}. What are the eigenvalues?

## Homework Equations

$$a_\alpha = \frac{1}{\sqrt{2\hbar}}(X_\alpha+ip_\alpha)$$

$$a_\alpha^\dagger=\frac{1}{sqrt{2\hbar}}(X_\alpha-iP_\alpha)$$

$$N_\alpha=a_\alpha^\dagger a_\alpha$$

$$J^a=\frac{\hbar}{2}\sum_{\alpha,\beta}a_\alpha^\dagger\sigma_{\alpha\beta}^aa_\beta$$

$$J^0=\frac{\hbar}{2}\sum_\alpha a_\alpha^\dagger a_\alpha$$

## The Attempt at a Solution

Ok, so I think that I already managed to get a) and c). I just put them her for the sake of completeness.

For points b) and e) I honestly have no idea where to get started.

And for point d) I started trying to substitute the definition of J

^{a}into the middle part of the equation. So I get $$\sum_a (J^a)^2=(J^1)^2+(J^2)^2+(J^3)^2 \\ = (\frac{\hbar}{2}\sum_{\alpha\beta}a_\alpha^\dagger \sigma^1 a_\beta)^2+(\frac{\hbar}{2}\sum_{\alpha\beta}a_\alpha^\dagger \sigma^2 a_\beta)^2+(\frac{\hbar}{2}\sum_{\alpha\beta}a_\alpha^\dagger \sigma^3 a_\beta)^2$$. But I don't know how to go on from here.

Any help would be appreciated.