Why Energy Eigenvalues Must Be Real
Inspired by a problem from PHYS 3316 (Basics of Quantum Mechanics) at Cornell University.
In quantum mechanics, the energy eigenvalues of the Schrödinger equation are always real numbers for normalizable states. This isn’t an axiom, but rather follows directly from demanding that probability is conserved.
Suppose an energy eigenvalue were complex: \(E = E_0 + i\Gamma\). A stationary state with this energy evolves as:
$$\Psi(x,t) = \psi(x)\,e^{-iEt/\hbar} = \psi(x)\,e^{-iE_0 t/\hbar}\,e^{\Gamma t/\hbar}$$
The first exponential is a phase – its modulus is 1, so it drops out when we compute the probability density \(|\Psi|^2\). The second term will remain. Normalization requires:
$$\int dx\,|\Psi|^2 = e^{2\Gamma t/\hbar}\int dx\,|\psi|^2 = 1$$
for all time. The integral \(\int|\psi|^2\) is a positive constant. For the product to equal 1 at all times, we need \(\Gamma = 0\). Therefore \(E\) is real.
If energy had an imaginary part, probability would either grow or decay exponentially – states would spontaneously appear or vanish. The real-valuedness of energy is a consequence of the conservation of probability, not an independent postulate.
It is worth unpacking what “normalization” actually means here, because the proof hinges on it. A state is normalizable when \(\int|\psi|^2\,dx\) converges to a finite number. Physically, this means the particle is somewhere – it may be spread over a large region, but not infinitely so. The probability of finding it somewhere in all of space is 1. A bound electron in a hydrogen atom is normalizable. A free electron described by a plane wave \(e^{ikx}\) is not – it extends uniformly across all space, and the integral diverges (although in this case, the state still has real energy).
So the proof is really saying: if a quantum state represents a persistent object with conserved total probability, then its energy must be real. The converse is where it gets interesting. In scattering theory and nuclear physics, we relax our requirements on the conservation of probability to allow for decaying objects. These unstable resonances are described by complex energies, where \(\Gamma\) is the decay width. Their probability does decay over time, which is the whole point. The proof does not forbid complex energies in general; it forbids them for states that persist.