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Schroedinger Equation into Two Linked Equations Involving Only Real Dependent Variables |
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The time-dependent Schroedinger equation is
where the wave function ψ is complex.
Let ψ(z, t)=φ(z, t)+iγ(z, t), where z denote the location vector and t is time, Then
The separation of the real and imaginary terms gives
Let Ψ denote the column vector with components φ and γ. The above system can be symbolically represented in matrix form as
where M is the matrix of operators
| 0 | H^/ |
| −H^/ | 0 | |
Let ψ=r·exp(iθ). Then
The RHS of Schroedinger's equation, H^(r·exp(iθ), cannot be separated at the general level because H^ usually involves the Laplacian operator which for a product of functions is complicated; i.e.,
Consider the case of a particle of mass m in a potential field V(z). Its Hamiltonian operator is
For one spatial dimension x
Then
Note that each term has a factor of exp(iθ). When the substitutions into the Schroedinger equation are made and the common factor of exp(iθ) cancelled the result is
A separation of real and imaginary terms gives
Now consider θ=Ω/h. This replacement results in the previous two equations becoming
The first of these reduces to
A nice touch is to let r=√ρ. Then ψψ*=ρ and thus ρ is the probability density function for the physical system. With this substitution
(To be continued.) .
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