Question

If a wavefunction of a one-dimensional particle in a box is given, how will we write it in Dirac notation using energy values as basis vectors?

Answer 1

You already have a solution, En and phi_n(x)?

Answer 2

Is phi(x,t) = alpha_n e^(-iE_nt/n) phi_n(x) what you are looking for? (quantization)

(E = hbar^2n^2pi^2/2ml^2, phi_n = sqrt(2/l) sin npix/l )

Answer 3

\[\Large \Sigma^{\infty}_{n=1}\,\, a_n|n\rangle \]

Answer 4

? @him1618

Answer 5

a sub n would be calculated how?

Answer 6

@him1618
As in any inner-product space, also in Hilbert space the expansion coeffs. in any basis which is orthonormal are easily calculated by Projections on each of the basis vectors.
We mean only the spatial dependence of the basis vectors and spatial inner product given by \[ \langle \Phi(x),\Psi(x) \rangle = \int_{\frak{R}} \Phi(x)^*\cdot\Psi(x) \,\,dx \]
If e.g. the basis vectors will be thos as brought above by @estudier
\[ \Large \phi(x,t) = \alpha_n e^{(-iE_nt/n)} \phi_n(x)\] where

Answer 7

\[ \phi_n = \sqrt(2/l) \sin(n\pi x/l) )\]

Answer 8

So if you have a solution (yet unknown!) for whcih you know some condition/conditions such as e.g. Total energy and/or boundary behavior and/or some Initial conditions and/or MOST COMMONLY: scattering asymptotic states at t=-infty and t=+infty then these conditions give you data to determine a_n -s

Answer 9

Boundary conditions phi(0) = phi(l) = 0

Answer 10

The alpha_n by setting t=0 and scalar product -> <phi_n | phi(t=0)>

Answer 11

got it..i was only confused for the coordinates a sub n thanks

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Answer 14

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Answer 23

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Answer 30

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Answer 31

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