# Probability for a non-hermitian hamiltonian

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1. Oct 25, 2015

### shinobi20

1. The problem statement, all variables and given/known data
Given a non-hermitian hamiltonian with V = (Re)V -i(Im)V. By deriving the conservation of probability, it can be shown that the total probability of finding a system/particle decreases exponentially as e(-2*ImV*t)/ħ

2. Relevant equations
Schrodinger Eqn, conservation of probability (∂P/∂t = -∇⋅j), P is the probability and j is the probability current density, Re V is the real part of V while Im V is the imaginary part of V

3. The attempt at a solution
It is known that ∫∫∫ P d3r = constant, that is ,the total probability for finding the particle should be constant throughout the universe.

iħ ∂ψ/∂t = -ħ2/2m ∇2ψ + Vψ = -ħ2/2m ∇2ψ + (Re)Vψ -i(Im)Vψ
By taking the complex conjugate, we have
-iħ ∂ψ*/∂t = -ħ2/2m ∇2ψ* +V*ψ* = -ħ2/2m ∇2ψ* + (Re)Vψ* +i(Im)Vψ*
By multiplying by ψ* the first equation and by ψ the second equation then subtracting the second from the first, we have
iħ ∂(ψψ*)/∂t = -ħ2/2m (ψ*∇2ψ-ψ∇2ψ*) - 2i (Im)V ψψ* = -∇⋅j - 2i (Im)V ψψ*

I don't know where to go already from here. Any suggestions or hints?

2. Oct 25, 2015

### blue_leaf77

Just use the definition of total probability in quantum mechanics: $\langle \psi(t) | \psi(t) \rangle$. Now $|\psi(t)\rangle = e^{-iVt/\hbar} |\psi(0)\rangle$. What will you get then when calculating $\langle \psi(t) | \psi(t) \rangle$?

3. Oct 25, 2015

### shinobi20

I will get ⟨ψ(t)|ψ(t)⟩ = ⟨ψ(0)|ψ(0)⟩ but what does this tell me? Also, I need to use the conservation of probability equation to derive the question.

4. Oct 25, 2015

### blue_leaf77

No, you won't because in this problem $V$ is not Hermitian. Try plugging in $V = \textrm{Re}[V] - i \textrm{Im}[V]$.

5. Oct 25, 2015

### shinobi20

Sorry, I forgot that it is not hermitian, so we will get e2i(Im)Vt

6. Oct 25, 2015

### blue_leaf77

It will be $\langle \psi(0) |e^{-i\textrm{Im}[V]t/\hbar}| \psi(0) \rangle$, at this point we should know how $\textrm{Im}[V]$ acts on $| \psi(0) \rangle$, i.e. whether it is a function of space at all. But since the question asks you to prove that "decreases exponentially as e(-2*ImV*t)/ħ", I think it is implied that $\textrm{Im}[V]$ is a constant and thus the sandwiched exponential operator may be taken out.

Last edited: Oct 25, 2015
7. Oct 25, 2015

### shinobi20

Oh! I did something wrong when taking the complex conjugate of V, but anyways, you really cleared everything. Thanks!

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