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the Time Independent Schrödinger Equation |
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Consider a particle of mass m moving a potential field V(x) in a one dimensional space. The Hamiltonian of this system is
where p is the particle's momentum.
Its time indepent Schrödinger equation is
where h is Planck's constant divided by 2π, ψ(x) is the system's wave
function and E is the system's energy.
Consider the Fourier transform of the equation
where Ψ(ω) and U(ω) are the Fourier transforms of ψ(x) and V(x), respectively. Ψ(ω) and U(ω) can be considered vectors in an infinite dimensional complex space. Let S be an infinite dimensional diagonal square matrix with ω² as the diagonal element of each row. Let W infinite dimensional square matrix with U(ω−ν) as the elements on each row. The above equation can then be represented as the operator (matrix) equation
where k is the scalar (½h²/m). Ψ is an eigenvector of (kS+W) and E is its eigenvalue.
The term kS can be considered a perturbation of the system represented by WΨ=EΨ.
If h→0 or m→∞ the quantum analysis should asymptotically approach the classical solution.
These limits correspond to k→0.
(To be continued.)
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