I have started learning quantum mechanics . I can now represent position eigenvector in momentum space(that is in the form of sum of momentum eigenvectors) and vice versa (fourier transforms of each other ). I can can represent energy eigenvector in position space(solving the time independent schrodinger equation) but stuck in doing the vice versa . I have tried but reached nowhere . Can someone explain me how can I solve the problem?
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For the oscillator, this is routine. But, in general, you are solving the TISE in position space... – Cosmas ZachosCommented Oct 20 at 20:19 -
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....here. – Cosmas ZachosCommented Oct 20 at 20:27
1 Answer
At your level, you might think of the eigenfunctions of your time-independent hermitean hamiltonian analogously to the exponential kernels of the Fourier transform,
|p⟩=∫dx eixp/ℏ√2πℏ|x⟩,
That is, if your orthonormal energy eigenfunctions are complete,
∑nψn(x)ψ∗n(y)=δ(x−y) ↔ ∑n|ψn⟩⟨ψn|=I,
To wit,
|ψn⟩=∫dx ψn(x)|x⟩ ↔ |x⟩=∑nψ∗n(x)|ψn⟩,
In any case, the complete orthonormal set of eigenfunctions of the hamiltonian are some type of transform you might analogize to the Fourier one. If it made you feel better, you might label the eigenvectors by |En⟩ instead of |ψn⟩.
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Sir , I have also got another way of solving this problem . We can first represent the position eigenvector in terms of momentum eigenvectors which are also kinetic energy eigenvectors (|p》=|(ke∗2m)^{.5}》 ). For a particular position at a time , potential energy is fixed so kinetic energy eigenvector is total energy or simply hamiltonian eigenvector . Am I correct ?– ReaderCommented Oct 21 at 19:09