Established physics · Quantum Mechanics 201 · Lesson 9 of 15

Bound states and quantized energy

Boundary conditions and the Hamiltonian determine allowed bound-state energies; quantum mechanics does not make every energy in every situation discrete.

Free science education from TRQC™ · Timeline-Relative Quantum Collapse™.

Picture it

Picture a wave confined between two fixed ends. Only certain standing-wave shapes fit the boundaries. A quantum particle confined by an ideal infinite well also has allowed stationary shapes, each with an allowed energy. The analogy helps you see the role of boundary conditions. The quantum wavefunction still predicts particle detections; it is not an ordinary vibrating string.

The allowed shapes must fit the problem

An infinite one-dimensional well confines a particle to an interval of length L, with the wavefunction zero at both ends. A standing wave with a whole number of half-wavelengths fits those conditions. Not every guessed shape is an energy eigenstate. More general initial states can be superpositions of the allowed eigenstates.

There is a lowest nonzero energy

In this ideal well, the lowest state has one half-wave across the interval. A spatially constant, nonzero function would fail the endpoint conditions, while the all-zero function cannot be normalized. The ground-state kinetic energy is therefore positive. This result belongs to the specified Hamiltonian and boundaries; an absolute zero of energy can otherwise be shifted by convention.

Level spacings depend on the system

For the infinite well, energies grow as n² and fall as 1/L². Doubling the well width divides each corresponding energy by four. Other systems have different spectra: a harmonic oscillator has equally spaced levels, while an atom has its own structure. “Quantized” does not mean one universal ladder spacing for all physical systems.

Bound states and continuum states coexist in physics

An isolated atom has discrete bound-state energies and also ionization/scattering possibilities. A free particle on an infinite line has a continuous momentum and energy spectrum. When an atom emits a photon between levels, the photon energy matches the energy difference, with the full process accounting for recoil and other relevant effects. Electrons are not little planets following the drawn probability lobes.

Constraints select the stationary shapes
  1. Specify a confining system
  2. Find the allowed wavefunctions
  3. Read its energy spectrum

The infinite-well example has n² level spacing; other Hamiltonians have other spectra.

Try the example on paper

At L = 1, E₁ = 1, E₂ = 4, E₃ = 9 in the displayed units. At L = 2, each becomes one quarter as large. The n-th state has n − 1 interior nodes.

Make it concrete

At L = 1, the first three scaled energies are 1, 4, and 9. The gap from n = 2 to n = 1 is 3, whereas the gap from n = 3 to n = 2 is 5. At L = 2, those energies become 1/4, 1, and 9/4. The n = 2 wavefunction has one interior node, where its position probability density is zero.

The precise version

For the infinite well on 0 < x < L, ψₙ(x) = √(2/L) sin(nπx/L), Eₙ = n²π²ℏ²/(2mL²), n = 1,2,… . There are n − 1 interior nodes. The demonstration displays energy in units E₁(L = 1), so its numerical energy is n²/L². A transition between levels releases or absorbs their energy difference.

Watch the trap

Do not turn a standing-wave sketch into a classical orbit, or assume that every quantum energy spectrum is discrete and equally spaced. Read the Hamiltonian and boundary conditions.

Keep this

Allowed stationary states come from the physical constraints. Their energy levels and probability shapes are linked.

Words for this lesson: bound state · quantization · eigenstate · stationary state · Hamiltonian · energy level

Check your understanding

Reason through each question before opening its answer and explanation.

1. Why are only certain stationary wave shapes allowed in the ideal well?

  1. Probability is allowed to vanish everywhere
  2. They must satisfy the Schrödinger equation and boundary conditions
  3. A person chooses an integer after observing the particle
Show answer and explanation

They must satisfy the Schrödinger equation and boundary conditions

The Hamiltonian and endpoint conditions select the allowed normalized eigenstates.

2. At L = 1 in the scaled units, what is E₂?

  1. 2
  2. 4
  3. 1/2
Show answer and explanation

4

The ideal-well energy is n²/L², so n = 2 gives 4.

3. If the well width doubles, what happens to a corresponding Eₙ?

  1. It becomes one quarter as large
  2. It always stays unchanged
  3. It becomes four times larger
Show answer and explanation

It becomes one quarter as large

The energy scales as 1/L² for fixed n and mass.

4. How many interior nodes does ψ₃ have?

  1. 3
  2. 0
  3. 2
Show answer and explanation

2

The n-th infinite-well eigenfunction has n − 1 interior nodes, excluding the two boundaries.

5. Are all quantum energy spectra discrete?

  1. No; free particles and scattering states can have continuous spectra
  2. Yes, every situation has the same ladder
  3. Only when no detector is present
Show answer and explanation

No; free particles and scattering states can have continuous spectra

Discreteness depends on the Hamiltonian, domain, and state class. Bound and continuum spectra are different possibilities.

6. What sets the photon energy in an idealized transition between stationary atomic levels?

  1. The larger level’s label alone
  2. The drawn orbital’s circumference
  3. The difference between the levels, with the full process accounting for recoil
Show answer and explanation

The difference between the levels, with the full process accounting for recoil

Energy conservation relates emitted or absorbed radiation to the level change and any other participating energy changes.

Further reading