Can a complex potential have real energies?

[eljose]

Let be the Hamiltonian:

[tex]H=T+V [/tex] where [tex]V(x)=A(x)+iB(x) [/tex]

then my question is if the Hamiltonian will have real energies..if we apply Ehrenfrest,s theorem for B:

[tex]i\hbar{\frac{d<B>}{dt}}=<[B,H]>[/tex]

then if B is a function of x and x and p do not commute the derivative of <B> can not be zero so its integral is also non-zero so we will always have that:

[tex]E=<H>=<T>+<A>+i<B> [/tex]

so the expected value of the Hamiltonian (the energy) will be complex.

 

[ZapperZ]

You already have not one, but TWO separate threads on this. Please finish up, or continue this business there!

https://www.physicsforums.com/showthread.php?t=85626
https://www.physicsforums.com/showthread.php?t=98118

Zz.

 

[denian]

Yes, a complex potential can result in a complex energy. In fact, this is a common occurrence in quantum mechanics. The Hamiltonian operator, which represents the total energy of a system, can have both real and imaginary components. This means that the energy of the system can also have both real and imaginary parts.

In classical mechanics, the Hamiltonian is always a real quantity. However, in quantum mechanics, the Hamiltonian operator can have complex eigenvalues and eigenfunctions. This is because in quantum mechanics, particles are described by wave functions which can have both real and imaginary parts.

In the case of a complex potential, the Hamiltonian will have both real and imaginary terms, resulting in a complex energy. This is perfectly acceptable in quantum mechanics and does not violate any laws of physics. In fact, complex potentials are often used in theoretical models to describe certain physical systems.

It is important to note that while the energy may be complex, the physical observables (such as position, momentum, and energy) will still be real. This is because in quantum mechanics, the expectation value of an observable is calculated by taking the inner product of the wave function with the corresponding operator. Since the wave function is a complex quantity, the inner product will result in a complex value, but the physical observable itself will still be real.

In summary, a complex potential can result in a complex energy in quantum mechanics, and this is a common occurrence. It does not violate any laws of physics and is a result of the mathematical framework of quantum mechanics.