Methods of separation

Assuming that the wave function (

Ψ (x, t)

) is separable, i.e. assuming the function of two variables can be written as the product of two different functions of a single variable:

Ψ (x, t) = ψ (x) T (t)

then, using standard mathematical techniques of Partial Differential equations, it can be shown that the wave equation can be rewritten as two distinct differential equations given by:

i ℏ \frac{1}{T (t)} \frac{d^{2} T (t)}{d t^{2}} = C

and

- \frac{ℏ^{2}}{2 m} \frac{d^{2} ψ (x)}{d x^{2}} + V (x) ψ (x) = C

where the first equation is solely dependent on time

T (t)

, and the second equation depends only on position

ψ (x)

, and where

C

is just a number. Generalizations can easily be made for higher numbers of spacial dimensions (up till

3

dimensions of space and

1

of time).

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Cyan Wind
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Why has KA not roll out a typesetting system such as LaTeX in comment section yet? We users surely need something to express math functions.
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