Question

The Hamiltonian of a harmonic oscillator is given by $H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2 = \hbar\omega(A^+A + \frac{1}{2})$ where $x = \sqrt{\frac{\hbar}{2m\omega}}(A^+ + A)$ $p = i\sqrt{\frac{m\hbar\omega}{2}}(A^+ - A)$ $A^+|u_n\rangle = \sqrt{n+1}|u_{n+1}\rangle$ $A|u_n\rangle = \sqrt{n}|u_{n-1}\rangle$ $[A, A^+] = 1$ and $E_n = \hbar\omega(n + \frac{1}{2}), n = 0, 1, 2,...$ where $|u_n(x)\rangle$ are real, normalized and orthogonal eigenfunctions of the harmonic oscillator. When a term $H' = \lambda x^3$ is added to the Hamiltonian H in which $\lambda$ is a real number, what is the effect of $H'$ (its the expectation value, $<u_n|H'|u_n>$)? Select one: A. $\frac{3}{2}\lambda\left(\frac{\hbar}{2m\omega}\right)^{3/2}$ B. $\frac{3}{2}\lambda\hbar\omega$ C. 0 D. $\lambda\left(\frac{\hbar}{2m\omega}\right)^{3/2}$ E. $n\lambda\left(\frac{\hbar}{2m\omega}\right)^{3/2}$ Your answer is correct. The correct answer is: 0

          The Hamiltonian of a harmonic oscillator is given by
$H = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2 = \hbar\omega(A^+A + \frac{1}{2})$
where
$x = \sqrt{\frac{\hbar}{2m\omega}}(A^+ + A)$
$p = i\sqrt{\frac{m\hbar\omega}{2}}(A^+ - A)$
$A^+|u_n\rangle = \sqrt{n+1}|u_{n+1}\rangle$
$A|u_n\rangle = \sqrt{n}|u_{n-1}\rangle$
$[A, A^+] = 1$ and $E_n = \hbar\omega(n + \frac{1}{2}), n = 0, 1, 2,...$
where $|u_n(x)\rangle$ are real, normalized and orthogonal eigenfunctions of the harmonic oscillator. When a term
$H' = \lambda x^3$ is added to the Hamiltonian H in which $\lambda$ is a real number, what is the effect of $H'$ (its the
expectation value, $<u_n|H'|u_n>$)?
Select one:
A. $\frac{3}{2}\lambda\left(\frac{\hbar}{2m\omega}\right)^{3/2}$
B. $\frac{3}{2}\lambda\hbar\omega$
C. 0
D. $\lambda\left(\frac{\hbar}{2m\omega}\right)^{3/2}$
E. $n\lambda\left(\frac{\hbar}{2m\omega}\right)^{3/2}$
Your answer is correct.
The correct answer is: 0

The Hamiltonian of a harmonic oscillator is given by
H = (p^2)/(2m) + (1)/(2)mω^2 x^2 = ħω(A^+A + (1)/(2))
where
x = √((ħ)/(2mω))(A^+ + A)
p = i√((mħω)/(2))(A^+ - A)
A^+|un⟩ = √(n+1)|un+1⟩
A|un⟩ = √(n)|un-1⟩
[A, A^+] = 1 and En = ħω(n + (1)/(2)), n = 0, 1, 2,...
where |un(x)⟩ are real, normalized and orthogonal eigenfunctions of the harmonic oscillator. When a term
H' = λ x^3 is added to the Hamiltonian H in which λ is a real number, what is the effect of H' (its the
expectation value, <un|H'|un>)?
Select one:
A. (3)/(2)λ((ħ)/(2mω))^3/2
B. (3)/(2)λħω
C. 0
D. λ((ħ)/(2mω))^3/2
E. nλ((ħ)/(2mω))^3/2
Your answer is correct.
The correct answer is: 0

Added by Kent G.

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