Virtual Labs

Quantum Mechanics of a Linear Harmonic Oscillator

1. What is the ground state energy of a quantum harmonic oscillator?

a: E₀ = 0 Explanation

Explanation

b: E₀ = ½ℏω (zero-point energy) Explanation

Explanation

c: E₀ = ℏω Explanation

Explanation

d: E₀ = 2ℏω Explanation

Explanation

2. What is the formula for the energy of the nth quantum state of a harmonic oscillator?

a: Eₙ = nℏω Explanation

Explanation

b: Eₙ = (n + ½)ℏω Explanation

Explanation

c: Eₙ = n²ℏω Explanation

Explanation

d: Eₙ = ℏω/n Explanation

Explanation

3. How many nodes does the wavefunction ψₙ(x) have for a quantum harmonic oscillator in state n?

a: 0 nodes for all states Explanation

Explanation

b: n nodes Explanation

Explanation

c: n + 1 nodes Explanation

Explanation

d: 2n nodes Explanation

Explanation

4. What determines the angular frequency ω of a harmonic oscillator?

a: Only the amplitude Explanation

Explanation

b: Only the energy level Explanation

Explanation

c: The spring constant k and mass m (ω = √(k/m)) Explanation

Explanation

d: Only the initial phase Explanation

Explanation

5. What is a 'coherent state' in quantum mechanics?

a: A state with zero energy Explanation

Explanation

b: A quantum state that most closely resembles classical harmonic oscillation Explanation

Explanation

c: A state with maximum uncertainty Explanation

Explanation

d: A state that never changes with time Explanation

Explanation

6. What type of polynomial functions appear in the harmonic oscillator wavefunctions?

a: Legendre polynomials Explanation

Explanation

b: Hermite polynomials Explanation

Explanation

c: Chebyshev polynomials Explanation

Explanation

d: Bessel polynomials Explanation

Explanation

7. In a quantum superposition of harmonic oscillator states, what does |cₙ|² represent?

a: The energy of state n Explanation

Explanation

b: The phase of state n Explanation

Explanation

c: The probability of measuring the system in state n Explanation

Explanation

d: The frequency of state n Explanation

Explanation

8. What is the spacing between adjacent energy levels in a quantum harmonic oscillator?

a: It decreases with increasing n Explanation

Explanation

b: It increases with increasing n Explanation

Explanation

c: It remains constant at ℏω Explanation

Explanation

d: It varies randomly Explanation

Explanation

9. What is the normalization condition for a quantum wavefunction?

a: The sum of all amplitudes must equal 1 Explanation

Explanation

b: The sum of all probability densities integrated over all space must equal 1 Explanation

Explanation

c: The maximum amplitude must be 1 Explanation

Explanation

d: The phase must be zero Explanation

Explanation

10. Why can't a quantum harmonic oscillator have zero energy?

a: Because of energy conservation Explanation

Explanation

b: Because of the Heisenberg uncertainty principle Explanation

Explanation

c: Because of the Pauli exclusion principle Explanation

Explanation

d: Because of relativistic effects Explanation

Explanation