File:SolveTimeIndepSchroedingerEqQuantumHarmonicOsc.gif
SolveTimeIndepSchroedingerEqQuantumHarmonicOsc.gif (434 × 268 pixels, file size: 1.16 MB, MIME type: image/gif, looped, 201 frames, 50 s)
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Description
[edit]General solutions of the time-independent 1D Schrödinger (differential) equation with the harmonic oscillator potential
This is an ordinary 2nd-order differential equation. Thus, for each value the set of solutions forms a vector space spanned by two linearly-independent basis functions. I have chosen the even-odd basis functions because the parity operator commutes with the Hamiltonian. The value is scanned in steps of 1/40 in units of .
The normalisation postulate of quantum mechanics requires us to select only those solutions that are normalisable, as these will be the only feasible results obtained from a measurement. Then, each normalised function (also called a Stationary_state or simply "standing wave") acquires the physical meaning of "probability amplitude" or Wave_function, and the associated eigenvalue acquires the meaning of "(eigen)energy". The rest of the solutions are unphysical, and thus discarded. Typically, this results in a discrete distribution of energies, which puts the "quantum" in quantum mechanics. In this case, the normalisation condition is equivalent to saying that the solutions must decay to 0 at infinity (technically known as the "boundary conditions").
For an analytical-yet-accessible derivation of the solutions see here.
The energy levels are non-degenerate (see Griffiths, for example).
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 21:11, 21 April 2024 | 434 × 268 (1.16 MB) | Rolancito (talk | contribs) | Make the E value clear. Increased number of frames for physical solutions | |
11:49, 20 April 2024 | 434 × 257 (1.07 MB) | Rolancito (talk | contribs) | Lower resolution such that thumbnail can be animated | ||
11:11, 20 April 2024 | 1,074 × 653 (2.78 MB) | Rolancito (talk | contribs) |
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