# Physical concept of imaginary energy [closed]

I want to know that what's the physical description of complex energy? The energies have the real and imaginary parts.

## closed as unclear what you're asking by John Rennie, Ben Crowell, Buzz, Jon Custer, YashasJun 2 at 8:30

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• In which context? The imaginary part of complex values in a physical context are usually a mathematical tool with no physical meaning. – Steeven May 23 at 14:02
• Energy is the expectation value of the Hamiltonian, and since this is a Hermitian operator the energy is always real. It cannot be complex. – John Rennie May 23 at 14:10
• Unstable states are those which have energy with an imaginary part. the complex potentials have complex energy . Is this correct? – Dana May 23 at 14:18
• What if Hamiltonian is not Hermitian? In the case of fractional quantum mechanics... – Dana May 23 at 14:26
• Please edit the question to explain more about the context. Is this classical? Quantum mechanical? If the latter, are these energies expectation values? – Ben Crowell May 23 at 17:09

## 2 Answers

The imaginary part of an energy tells you the lifetime of a state. The amplitude to detect a particle at time t with a complex mass $$m=m_R+im_I$$ evolves in time as

$$e^{i\frac{mt}{\hbar}}=e^{i\frac{(m_R+im_I)t}{\hbar}}=e^{i\frac{m_R t}{\hbar}}e^{\frac{-m_I t}{\hbar}}$$ Therefore, the lifetime of the particle is $$\tau=\frac{\hbar}{m_I}$$. Another example, in colliding $$e^+e^- \rightarrow \psi \rightarrow e^+e^-$$ the scattering amplitude as a function of the center of mass energy $$E_{cm}$$ of the $$e^+e^-$$ has a pole $$ScatteringAmplitude \approx \frac {1}{E_{cm}-m_{\psi}}$$ In the complex energy plane $$E_{cm}$$ moves along the real axis and the complex mass $$m_{\psi}$$ is above the axis because of its imaginary part. The farther $$m_{\psi}$$ is above the real axis, the broader the resonance is as $$E_{cm}$$ moves by, and the shorter the lifetime of the particle $$\psi$$.

As @John Rennie commented, the energy eigenvalues of a Hermitian Hamiltonian H are real. The H is Hermitian because $$e^{i\frac{Ht}{\hbar}}$$ is unitary (ie: the sum of the probabilities to be in some eigenstate always remains 1). An eigenstate of a Hermitian H does not go away with time. The particle $$\psi$$ with complex mass is not an eigenstate of this Hermitian H.

By definition in both classical and quantum mechanics the energy cannot be a complex number.

In classical mechanics, given a Lagrangian $$L = T-V$$, the energy is defined as $$E_{L} = v\frac{\partial L}{\partial v} - L$$ one could make an argument for the Lagrangian to not necessarily be the difference between kinetic and potential term; nevertheless it will always be a quadratic form of real functions of position and velocities because the equations of motion must be of second order.

In quantum mechanics the energy is the expectation value of a Hermitian operator, therefore it is a real number. The original operator you take the expectation value of must by all means by Hermitian due to the conservation of the norm for states in quantum mechanics (or conservation of probability, or however else you want to call it).