OpenStudy (anonymous):

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?

5 years ago
OpenStudy (anonymous):

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

5 years ago
OpenStudy (anonymous):

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 )

5 years ago
OpenStudy (anonymous):

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

5 years ago
OpenStudy (anonymous):

? @him1618

5 years ago
OpenStudy (anonymous):

a sub n would be calculated how?

5 years ago
OpenStudy (anonymous):

@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

5 years ago
OpenStudy (anonymous):

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

5 years ago
OpenStudy (anonymous):

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

5 years ago
OpenStudy (anonymous):

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

5 years ago
OpenStudy (anonymous):

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

5 years ago
OpenStudy (anonymous):

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

5 years ago
OpenStudy (anonymous):

Are you a "green new user" concerning the customs of this humble abode ?

5 years ago