# How can one calculate the phase difference between two quantum harmonic oscillator (Hermite-Gauss) states?

The analytic solutions of a quantum harmonic oscillator are given by Hermite-Gauss states, which differ in the order $n$ of the Hermite polynomials. If two such states are plotted, there will be a relative phase shift. How can that be calculated analytically?

• I try to understand your question. Is the phase shift time dependent or time independent? Commented Jun 28, 2014 at 4:48

According to quantum mechanics postulates phase ratio has no physical meaning. It has significance only for interference phenomena which occure to the waves which can spread accross the space. Pure harmonic oscillator states are bound ones. They can not interfere. However, Hermite-states phase ratio is given by general formula exp(-iEt2Π/h) where E is the energy of the n-th state. To describe entanglement, one is to use two different particles and two different states, e.g. n-th and m-th Hermite-states. Typical entangled state is proportional to Ψ(1)Φ(2)± Ψ(2)Φ(1), where Ψ and Φ denotes different states and 1,2 different particles. Using only Hermite states gives {Hn(y)Hm(x)±Hm(y)Hn(x)}×exp{-i2πt(En+Em)/h}. The exponent of its phase ratio is proportional to the sum of their energy. The locations of the particles are described by x ( three dimensions ) and y ( similarly). Because we are talking about nonrelativistic quantum mechanics the time of each particle, according to Galileo transformation , is the same.

Hermite-Gauss states are not waves, they are only static amplitudes of probability. Hermite polynomials have no phase at all. The phase of harmonic oscillator states depends only on time. I thought that used word "plotted" had been a mistake, instead of using "entangled".

• I was not talking about interference. If we plot two different Hermite states, there would be a relative phase shift between the two. Would that be spatial or temporal and how can that be analytically calculated if possible?
– Ana
Commented Jun 30, 2014 at 19:34