# 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.

• 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 '19 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 '19 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 '19 at 14:18
• What if Hamiltonian is not Hermitian? In the case of fractional quantum mechanics... – Dana May 23 '19 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? – user4552 May 23 '19 at 17:09

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.
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.