Question 1 a) Write down the one-dimensional time-dependent Schro ̈dinger equation, for a particle de- scribed by a wavefunction Ψ(t, x) in a potential V (t, x). b) State the condition that the potential must obey in order to derive the time-independent Schro ̈dinger equation from the time-dependent equation. c) Write down a mathematical expression for the wavefunction of a particle in an energy eigen- state of such a potential. Your answer should be given in terms of the spatially dependent part of the wavefunction, ψ(x), and a factor that depends on the energy eigenvalue of the particle, E. d) Using your answers to parts a)-c) of this question, derive the one-dimensional time-independent Schro ̈dinger equation for the particle described by the wavefunction ψ(x).
Question 1
a) Write down the one-dimensional time-dependent Schro ̈dinger equation, for a particle de-
scribed by a wavefunction Ψ(t, x) in a potential V (t, x).
b) State the condition that the potential must obey in order to derive the time-independent
Schro ̈dinger equation from the time-dependent equation.
c) Write down a mathematical expression for the wavefunction of a particle in an energy eigen- state of such a potential. Your answer should be given in terms of the spatially dependent part of the wavefunction, ψ(x), and a factor that depends on the energy eigenvalue of the particle, E.
d) Using your answers to parts a)-c) of this question, derive the one-dimensional time-independent Schro ̈dinger equation for the particle described by the wavefunction ψ(x).
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