Separation of variables to solve Schrodinger equations


by ace1719
Tags: quantum physics, schrodinger equation, separation of variab
ace1719
ace1719 is offline
#1
Apr12-13, 12:49 PM
P: 11
I've found many articles online that explain how to solve the Schrodinger equation for a potential dependent on x, but not for one dependent on t. A couple articles said that you could not use separation of variables to solve the Schrodinger equation with a time dependent potential, but they did not explain why. Why can you not use separation of variables to solve the Schrodinger equation with a time dependent potential, specifically; V(t)=A*cos(ωt), where A is a constant potential and ω is the angular frequency. Thanks!
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tadchem
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#2
Apr12-13, 01:42 PM
P: 61
Technically, a time-dependent field is not a 'potential' field. The S equation is soluble by separation of variables because the equation can be arranged so that the time-dependence is entirely one one side of the = sign. The other side of the = sign has no time-dependence. Mathematically, this can happen if and only if both the expressions are equal to a constant. In physical terms this constant works out to be the total energy. The wave represents a solution to an oscillator system.
If a time-dependent field is present, the solution is time-dependent and represents a *driven* oscillator, with energy being exchanged between the system and the environment. The source/sink of this variable energy is the time-dependent field.
Specifically, if the time-dependent potential is applied as an operator to a spatially-dependent wave, the solution to the S Equation would require Green's Functions, and the orthogonal coordinates needed for separating the variables would be mixed coordinates in space and time. They would represent nodes in the space/time-dependent wave solution.
ace1719
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#3
Apr14-13, 02:51 PM
P: 11
Quote Quote by tadchem View Post
Technically, a time-dependent field is not a 'potential' field. The S equation is soluble by separation of variables because the equation can be arranged so that the time-dependence is entirely one one side of the = sign. The other side of the = sign has no time-dependence. Mathematically, this can happen if and only if both the expressions are equal to a constant. In physical terms this constant works out to be the total energy. The wave represents a solution to an oscillator system.
If a time-dependent field is present, the solution is time-dependent and represents a *driven* oscillator, with energy being exchanged between the system and the environment. The source/sink of this variable energy is the time-dependent field.
Specifically, if the time-dependent potential is applied as an operator to a spatially-dependent wave, the solution to the S Equation would require Green's Functions, and the orthogonal coordinates needed for separating the variables would be mixed coordinates in space and time. They would represent nodes in the space/time-dependent wave solution.
Thanks a lot for the explanation, but just to clarify, is there nothing mathematically wrong with doing separation of variables with a time dependent potential?

tadchem
tadchem is offline
#4
Apr15-13, 07:08 AM
P: 61

Separation of variables to solve Schrodinger equations


Mathematically, separation of variables works best when the variables are orthogonal. In static problems Time is always orthogonal to the spatial variables.
In dynamic problems (time-dependent) the separability occurs when the chosen coordinates represent represent independent modes of vibration. Each vibrational mode becomes a coordinate with its own potential and wave solutions.
Chemists who study vibrational spectra of individual molecules (usually infrared) work with this daily. Sometimes two modes of vibration have the same symmetry and frequency ω , such as the 'bending' modes of a CO2 molecule. In that case the vibrations define a 'subspace' of fewer dimensions that is completely separable from the remaining modes, but which are indistinguishable from each other. Herzberg is a good starting point for electronic spectra of small molecules.
DrDu
DrDu is offline
#5
Apr15-13, 07:29 AM
Sci Advisor
P: 3,375
Why don't you just try it out?


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