Virtual Labs

Quantum Mechanics of a Particle in a Potential Well

1. What does the quantum number n represent in the particle-in-a-box model?

a: The amplitude of the wavefunction Explanation

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b: The energy level of the particle Explanation

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c: The length of the box Explanation

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d: The time interval of oscillation Explanation

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2. What is the time-dependent wavefunction for a particle in a 1D box?

a: Ψ(x,t) = A·cos(nπx/L)·sin(ωt) Explanation

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b: Ψ(x,t) = A·sin(nπx/L)·cos(ωt) Explanation

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c: Ψ(x,t) = A·sin(nx/L)·cos(t) Explanation

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d: Ψ(x,t) = A·cos(nx/L)·sin(t) Explanation

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3. What happens to the energy levels when the box length L decreases?

a: Energy levels move closer together Explanation

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b: Energy levels remain unchanged Explanation

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c: Energy levels become more widely spaced (higher energy) Explanation

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d: Energy levels become continuous Explanation

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4. What does |Ψ|² represent in quantum mechanics?

a: The energy of the particle Explanation

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b: The momentum of the particle Explanation

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c: The probability density of finding the particle Explanation

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d: The velocity of the particle Explanation

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5. How many nodes does a wavefunction have for quantum number n?

a: n nodes Explanation

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b: n + 1 nodes Explanation

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c: n - 1 nodes Explanation

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d: 2n nodes Explanation

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6. What is the boundary condition for the wavefunction at the walls of an infinite potential well?

a: Ψ = maximum at boundaries Explanation

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b: Ψ = 0 at boundaries Explanation

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c: Ψ = 1 at boundaries Explanation

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d: Ψ is undefined at boundaries Explanation

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7. What is the formula for energy levels in a 1D infinite potential well?

a: Eₙ = nh/(8mL²) Explanation

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b: Eₙ = n²h²/(8mL²) Explanation

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c: Eₙ = nh²/(8mL) Explanation

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d: Eₙ = n²h/(8mL) Explanation

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8. Why can't a particle in an infinite potential well have zero energy (E = 0)?

a: Because zero energy violates the Heisenberg uncertainty principle Explanation

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b: Because the particle has too much mass Explanation

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c: Because the box width is infinite Explanation

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d: Because the wavefunction becomes imaginary Explanation

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9. What is the de Broglie wavelength (λ) for a particle in the nth state of a box of length L?

a: λ = L/n Explanation

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b: λ = 2L/n Explanation

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c: λ = nL/2 Explanation

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d: λ = L/2n Explanation

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10. What is the normalized spatial wavefunction for a particle in a 1D box?

a: Ψₙ(x) = sin(nπx/L) Explanation

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b: Ψₙ(x) = √(1/L)·sin(nπx/L) Explanation

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c: Ψₙ(x) = √(2/L)·sin(nπx/L) Explanation

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d: Ψₙ(x) = 2·sin(nπx/L) Explanation

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