 Energy eigenspace

ID: energy-eigenspace

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Energy eigenspace by

Codex  0 Created 2026-09-24 Updated 2026-10-14

The energy eigenspace at

E

is the vector subspace

E_{E} = ker (H - E I) = {ψ \in D (H) : H ψ = E ψ},

where

H

is the Hamiltonian operator and

I

is the identity operator. It is nonzero exactly when

E

is an energy eigenvalue. Its dimension is the multiplicity of that energy eigenvalue; degenerate energy levels have multiplicity greater than one. The ground-state subspace is

E_{E_{0}}

, and its dimension is the ground-state degeneracy. A vector in an energy eigenspace need not be one of a specially chosen orthonormal basis of energy eigenstates.

Linearity makes the useful point immediate: if

H ϕ_{1} = E ϕ_{1}

and

H ϕ_{2} = E ϕ_{2}

, then

H (a ϕ_{1} + b ϕ_{2}) = E (a ϕ_{1} + b ϕ_{2})

. Every normalized nonzero combination stays in the same energy eigenspace. For a time-independent Hamiltonian operator, its evolution is multiplication by the common global phase

e^{- i Et /ℏ}

, so it is a stationary state. A quantum superposition within a degenerate energy eigenspace can change spatial shape when its coefficients are chosen differently, but its density does not subsequently beat under this unchanged Hamiltonian operator.

Across distinct energy eigenspaces, relative quantum phases evolve. For orthonormal eigenfunctions

ϕ_{1}, ϕ_{2}

of the Hamiltonian operator with energy eigenvalues

E_{1} \neq = E_{2}

, an equal-weight quantum superposition has

ψ (x, t) = \frac{e ^{- i E_{1} t /ℏ} ϕ _{1} ( x ) + e ^{- i E_{2} t /ℏ} ϕ _{2} ( x )}{2},

and position probability density

∣ ψ (x, t) ∣^{2} = \frac{∣ ϕ _{1} ( x ) ∣ ^{2} + ∣ ϕ _{2} ( x ) ∣ ^{2}}{2} + Re [ϕ_{1} (x)^{*} ϕ_{2} (x) e^{- i (E_{2} - E_{1}) t /ℏ}] .

The interference term can change local probabilities even though the Born rule assigns the same time-independent probability

1/2

to each energy measurement outcome. Orthogonality makes its integral zero, preserving wavefunction normalization. This pure state is not a stationary state; some particular measurement distributions may nevertheless remain constant. An incoherent mixed state with the same energy populations has no such interference term.

Figure 1.
An excited energy eigenstate can be stationary while a two-energy superposition beats
. Exact infinite square well densities, not measured data, with $u = x / L$ and $θ = (E_{2} - E_{1}) t /ℏ$ . The left panel shows the unchanged density of the second energy eigenstate at three times. The right panel shows a coherent equal-weight quantum superposition at $θ = 0, π /2, π$ ; its energy populations remain $1/2, 1/2$ . The dashed curve is the density of the corresponding incoherent mixed state. Details.

For the pictured infinite square well,

θ = 3 E_{1} t /ℏ

and the interference repeats after

T = 2 π ℏ/ (3 E_{1})

. At

t = 0

, direct integration gives

⟨ x ⟩ = L /2 - 16 L / (9 π^{2})

; halfway through the period it is

L /2 + 16 L / (9 π^{2})

. This is a moving expectation value, rather than a trajectory of an individual particle. The distinction between an energy eigenspace and a combination of different energy eigenspaces is what makes the different behavior possible.

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