Question

3. A harmonic oscillator in two-dimensions can be written $H = H_1 + H_2$ where $H_1 = \frac{1}{2m}p_x^2 + \frac{1}{2}m\omega^2 x^2$ The angular momentum $L = \mathbf{r} \times \mathbf{p}$ is $L = x p_y - y p_x$ Define $A = \frac{1}{2\omega}(H_1 - H_2)$ $B = \frac{L}{2}$ $C = -i[A, B]/\hbar$ a) Give the explicit form of the operator C. b) Show that A, B, C, H are Hermitian operators. c) Show that the set A, B, C are closed under commutation, i.e., $[A, C] = i\hbar B$ $[B, C] = i\hbar A$ d) Show that $A^2 + B^2 + C^2 = H^2/4\omega^2 + \text{constant}$. What is the constant? e) Show that A, B, C each commute with H.

          3. A harmonic oscillator in two-dimensions can be written
$H = H_1 + H_2$
where
$H_1 = \frac{1}{2m}p_x^2 + \frac{1}{2}m\omega^2 x^2$
The angular momentum $L = \mathbf{r} \times \mathbf{p}$ is
$L = x p_y - y p_x$
Define
$A = \frac{1}{2\omega}(H_1 - H_2)$
$B = \frac{L}{2}$
$C = -i[A, B]/\hbar$
a) Give the explicit form of the operator C.
b) Show that A, B, C, H are Hermitian operators.
c) Show that the set A, B, C are closed under commutation, i.e.,
$[A, C] = i\hbar B$
$[B, C] = i\hbar A$
d) Show that
$A^2 + B^2 + C^2 = H^2/4\omega^2 + \text{constant}$.
What is the constant?
e) Show that A, B, C each commute with H.

3. A harmonic oscillator in two-dimensions can be written
H = H1 + H2
where
H1 = (1)/(2m)px^2 + (1)/(2)mω^2 x^2
The angular momentum L = 𝐫×𝐩 is
L = x py - y px
Define
A = (1)/(2ω)(H1 - H2)
B = (L)/(2)
C = -i[A, B]/ħ
a) Give the explicit form of the operator C.
b) Show that A, B, C, H are Hermitian operators.
c) Show that the set A, B, C are closed under commutation, i.e.,
[A, C] = iħ B
[B, C] = iħ A
d) Show that
A^2 + B^2 + C^2 = H^2/4ω^2 + constant.
What is the constant?
e) Show that A, B, C each commute with H.

Added by Bego-A C.

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