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).