a) Write down the one-dimensional time-dependent Schrödinger equation, for a particle de- scribed by a wave function (x,t) in a potential V(x,t). b) State the condition that the potential must obey in order to derive the time-independent Schrödinger 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, (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 Schrödinger equation for the particle described by the wavefunction (x).
a) Write down the one-dimensional time-dependent Schrödinger equation, for a particle de- scribed by a wave function (x,t) in a potential V(x,t). b) State the condition that the potential must obey in order to derive the time-independent Schrödinger 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, (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 Schrödinger equation for the particle described by the wavefunction (x).
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[Solution for Part A in second image - do C, B and D]
Transcribed Image Text:a)
b)
Write down the
one-dimensional time-dependent Schrödinger equation, for a particle de-
scribed by a wavefunction (x,t) in a potential V(x,t).
State the condition that the potential must obey in order to derive the time-independent
Schrödinger 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, (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
Schrödinger equation for the particle described by the wavefunction (™).
Transcribed Image Text:ih
24(x, t)
at
ħ² a²(x, t)
2m əx²
·+V(x, t)(x, t)
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