Questions: Time-Independent Schrödinger Equation and Eigenvalues

Question 1

Why do bound quantum systems have discrete energy levels rather than a continuous range of allowed energies?

• A. The Heisenberg uncertainty principle directly prohibits energies between allowed values
• B. Physicists assume quantization and build it into the Schrödinger equation as a postulate
• C. Normalizable solutions satisfying physical boundary conditions exist only for a discrete set of eigenvalues
• D. The wave function collapses to discrete values during measurement

Answer: (C)

Quantization is not assumed — it falls out of the mathematics. The time-independent Schrödinger equation Ĥφ = Eφ has solutions for any energy E, but most of those solutions blow up at infinity or fail to satisfy the boundary conditions (e.g., φ = 0 at hard walls, φ → 0 as r → ∞). Only for a discrete set of energies Eₙ do normalizable solutions exist. Between the allowed energies, there are simply no physically acceptable solutions. This is the deep reason quantization is real and not just a modeling assumption.

Question 2

A particle is in stationary state Ψ(x,t) = φₙ(x)e^(−iEₙt/ℏ). What does the probability density |Ψ(x,t)|² look like at two different times t₁ and t₂?

• A. It oscillates between two different distributions as the phase rotates
• B. It is identical at both times — the probability distribution does not change
• C. It slowly spreads out over time as the wave function disperses
• D. It collapses to a point when the energy is measured at t₂

Answer: (B)

|Ψ(x,t)|² = |φₙ(x)e^(−iEₙt/ℏ)|² = |φₙ(x)|²·|e^(−iEₙt/ℏ)|² = |φₙ(x)|², because the magnitude of any complex exponential e^(iθ) is 1. The time-dependent phase factor cancels when you take the modulus squared, leaving a probability distribution that is identical at every time. 'Stationary' means the probabilities don't change — not that the particle is frozen or that the wave function isn't oscillating.

Question 3

Energy quantization in bound quantum systems is a physical assumption that should be imposed on the Schrödinger equation from outside.

• T. True
• F. False

Answer: False

This is the key insight: quantization is a mathematical consequence, not an additional physical assumption. When you solve Ĥφ = Eφ and require that solutions be normalizable and satisfy boundary conditions, you find that acceptable solutions exist only for a discrete set of energies. The mathematics forces quantization on you. This is why the eigenvalue approach is so powerful — it derives the energy spectrum rather than assuming it.

Question 4

In a stationary state, measuring the particle's energy multiple times always yields the same result: the eigenvalue Eₙ.

• T. True
• F. False

Answer: True

A stationary state is by definition an eigenstate of the Hamiltonian with eigenvalue Eₙ. From the measurement postulate of quantum mechanics, measuring an observable on an eigenstate of that observable always returns the eigenvalue with certainty. This is in contrast to a superposition state, where energy measurements yield different eigenvalues probabilistically. Stationary states have perfectly definite energy — zero uncertainty in energy, consistent with the energy-time uncertainty relation (a stationary state has a definite frequency and therefore a definite energy).

Question 5

Explain why a 'stationary state' does not mean the particle is at rest or that nothing is happening physically.

Model answer: A stationary state is one where the probability distribution |Ψ(x,t)|² does not change over time. The wave function itself is still evolving — it picks up the phase factor e^(−iEₙt/ℏ), which oscillates in time. The particle is still moving (it has kinetic energy), but the probability of finding it in any region remains constant. 'Stationary' refers to time-independence of probabilities, not of the wave function or the particle's motion.

The confusion arises because 'stationary' sounds like 'stopped.' But the Schrödinger equation shows the wave function always evolves as Ψ(x,t) = φ(x)e^(−iEt/ℏ). In a stationary state, this phase factor is global — it multiplies the entire wave function — so when you compute |Ψ|², it cancels. If the particle were actually at rest, it would have zero momentum and zero kinetic energy, which the uncertainty principle rules out for confined particles.