Chapter 4 Quiz: The Quantum Harmonic Oscillator

Multiple Choice

Q1. The energy levels of the quantum harmonic oscillator are:

(a) $E_n = n\hbar\omega$ for $n = 0, 1, 2, \ldots$ (b) $E_n = (n + \frac{1}{2})\hbar\omega$ for $n = 0, 1, 2, \ldots$ (c) $E_n = (n + \frac{1}{2})^2\hbar\omega$ for $n = 0, 1, 2, \ldots$ (d) $E_n = n^2\hbar^2\pi^2/(2mL^2)$ for $n = 1, 2, 3, \ldots$

Q2. The ground state energy of the QHO is nonzero because:

(a) The potential energy is defined to be positive (b) The Heisenberg uncertainty principle prevents the particle from being simultaneously at rest and at the potential minimum (c) The particle has nonzero mass (d) Energy conservation requires a minimum kinetic energy

Q3. The spacing between adjacent energy levels of the QHO is:

(a) Increasing with $n$ (b) Decreasing with $n$ (c) Constant, equal to $\hbar\omega$ (d) Constant, equal to $\frac{1}{2}\hbar\omega$

Q4. The wavefunction $\psi_4(x)$ of the QHO has:

(a) 3 nodes and even parity (b) 4 nodes and even parity (c) 4 nodes and odd parity (d) 5 nodes and even parity

Q5. The raising (creation) operator $\hat{a}^\dagger$ acting on the state $|n\rangle$ gives:

(a) $\sqrt{n}\,|n+1\rangle$ (b) $\sqrt{n+1}\,|n+1\rangle$ (c) $\sqrt{n}\,|n-1\rangle$ (d) $(n+1)|n+1\rangle$

Q6. The fundamental commutation relation for the ladder operators is $[\hat{a}, \hat{a}^\dagger] = 1$. This relation is a direct consequence of:

(a) The orthogonality of Hermite polynomials (b) The canonical commutation relation $[\hat{x}, \hat{p}] = i\hbar$ (c) The normalization of wavefunctions (d) Energy conservation

Q7. Which of the following is TRUE about coherent states $|\alpha\rangle$?

(a) They are eigenstates of the number operator $\hat{n}$ (b) They are eigenstates of the annihilation operator $\hat{a}$ (c) They are eigenstates of the Hamiltonian $\hat{H}$ (d) They have a definite number of quanta

Q8. A coherent state $|\alpha\rangle$ with $|\alpha|^2 = 9$ has a photon number distribution that is:

(a) Peaked sharply at $n = 9$ with no spread (b) A Poisson distribution with mean 9 and standard deviation 3 (c) A Gaussian distribution with mean 9 and standard deviation 3 (d) Uniformly distributed between $n = 0$ and $n = 18$

Q9. The quantum harmonic oscillator appears universally in physics primarily because:

(a) Most potentials are exactly parabolic (b) The Taylor expansion of any smooth potential around a stable minimum is quadratic to lowest nontrivial order (c) Planck assumed oscillators in blackbody radiation (d) Spring forces are the most common force in nature

Q10. In the algebraic method, the proof that the eigenvalues of $\hat{n} = \hat{a}^\dagger\hat{a}$ are non-negative relies on:

(a) The Hermite polynomials having real zeros (b) The norm of any vector being non-negative: $\|\hat{a}|n\rangle\|^2 = n \geq 0$ (c) The commutator $[\hat{a}, \hat{a}^\dagger]$ being positive (d) The potential energy being non-negative

Q11. The Hermite polynomial $H_3(\xi)$ is:

(a) $8\xi^3 - 12\xi$ (b) $8\xi^3 + 12\xi$ (c) $4\xi^3 - 2\xi$ (d) $6\xi^3 - 6\xi$

Q12. Which statement best describes the relationship between the analytical and algebraic methods for solving the QHO?

(a) The analytical method is correct; the algebraic method is an approximation (b) They give the same energy spectrum but the algebraic method cannot produce wavefunctions (c) They give the same results; the analytical method gives explicit wavefunctions while the algebraic method reveals the operator structure (d) The algebraic method is more general because it works for any potential

True/False

Q13. True or False: The quantum harmonic oscillator has a finite number of bound states.

Q14. True or False: The probability of finding a QHO in its ground state beyond the classical turning point is exactly zero.

Q15. True or False: Coherent states remain coherent states under time evolution in a harmonic potential.

Q16. True or False: The zero-point energy $E_0 = \frac{1}{2}\hbar\omega$ means we can extract energy from the vacuum of the electromagnetic field.

Short Answer

Q17. The lowering operator acts on the ground state as $\hat{a}|0\rangle = ?$. Explain in one sentence why this result is physically necessary.

Q18. State the Casimir effect in two sentences and explain its connection to zero-point energy.

Q19. A diatomic molecule with reduced mass $\mu$ has a vibrational frequency $\omega$. Write the expression for the vibrational energy levels and calculate the zero-point energy if $\omega = 5.0 \times 10^{14}$ rad/s.

Q20. In quantum field theory, each mode of the electromagnetic field is treated as a quantum harmonic oscillator. What are the "quanta" of these oscillators called, and what do the creation and annihilation operators $\hat{a}^\dagger$ and $\hat{a}$ physically represent?

Answer Key

Q1: (b) — The energy levels are $E_n = (n + \frac{1}{2})\hbar\omega$ with $n = 0, 1, 2, \ldots$

Q2: (b) — The uncertainty principle prevents simultaneous localization in position and momentum, forcing a minimum energy.

Q3: (c) — The spacing is constant: $E_{n+1} - E_n = \hbar\omega$ for all $n$.

Q4: (b) — $\psi_n$ has $n$ nodes and parity $(-1)^n$. For $n = 4$: 4 nodes, even parity.

Q5: (b) — $\hat{a}^\dagger|n\rangle = \sqrt{n+1}\,|n+1\rangle$.

Q6: (b) — $[\hat{a}, \hat{a}^\dagger] = 1$ follows directly from $[\hat{x}, \hat{p}] = i\hbar$ and the definitions of $\hat{a}$, $\hat{a}^\dagger$.

Q7: (b) — Coherent states are defined as eigenstates of $\hat{a}$: $\hat{a}|\alpha\rangle = \alpha|\alpha\rangle$.

Q8: (b) — The photon number distribution is Poissonian with mean $\bar{n} = |\alpha|^2 = 9$ and $\Delta n = \sqrt{\bar{n}} = 3$.

Q9: (b) — The Taylor expansion guarantees that any stable equilibrium looks like a harmonic oscillator for small oscillations.

Q10: (b) — Since $\langle n|\hat{a}^\dagger\hat{a}|n\rangle = \|\hat{a}|n\rangle\|^2 \geq 0$, the eigenvalue $n$ must be non-negative.

Q11: (a) — $H_3(\xi) = 8\xi^3 - 12\xi$ (physicist's convention).

Q12: (c) — Both methods give identical results. The analytical method produces explicit Hermite polynomial wavefunctions; the algebraic method reveals the ladder operator structure and generalizes to other systems.

Q13: False — The QHO has infinitely many bound states: $|0\rangle, |1\rangle, |2\rangle, \ldots$ with energies increasing without bound.

Q14: False — There is approximately a 16% probability of finding the ground-state particle beyond the classical turning points (classically forbidden region).

Q15: True — Under harmonic evolution, $|\alpha\rangle$ evolves to $|e^{-i\omega t}\alpha\rangle$, which is still a coherent state (with a rotated parameter).

Q16: False — The vacuum is the lowest energy state by definition. Zero-point energy is real (Casimir effect), but it cannot be "extracted" as usable energy because there is no lower state to transition to.

Q17: $\hat{a}|0\rangle = 0$ (the zero vector). This is physically necessary because the ground state has the lowest possible energy; there is no state below it for the lowering operator to reach.

Q18: The Casimir effect is an attractive force between two uncharged parallel conducting plates in vacuum, caused by the restriction of electromagnetic vacuum modes (and their zero-point energies) between the plates. The difference in zero-point energy density between the region inside and outside the plates produces a measurable force, confirming that zero-point fluctuations are physically real.

Q19: $E_v = (v + \frac{1}{2})\hbar\omega$ for $v = 0, 1, 2, \ldots$. Zero-point energy: $E_0 = \frac{1}{2}\hbar\omega = \frac{1}{2}(1.055 \times 10^{-34})(5.0 \times 10^{14}) \approx 2.6 \times 10^{-20}$ J $\approx 0.16$ eV.

Q20: The quanta are called photons. The creation operator $\hat{a}^\dagger$ adds one photon to the mode (photon emission), and the annihilation operator $\hat{a}$ removes one photon from the mode (photon absorption).