A quantum mechanical particle of mass m moves in a 1 D potential where a) Estimate the ground state energy of the particle. b) Sketch the wave function to the best of your ability.

A particle is described by the wavefunction below, where L = 1 nanometer. a) Determine the value of the constant c. b) Draw the graphs of the wavefunction and the probability density. c) Calculate the probability of finding the particle in the region x > 1 nanometer

For a particle in a box, what would the probability distribution function Ic I2 look like if the particle behaved like a classical (Newtonian) particle? Do the actual probability distributions approach this classical form when n is very large? Explain.

a)  Write down the one-dimensional time-dependent Schro ̈dinger equation for a wavefunction Ψ(t, x) in a potential V (x). b)  Write down the one-dimensional time-independent Schro ̈dinger equation for a wavefunc- tion ψ(x) in a potential V (x). c)  Assuming that Ψ(t,x) corresponds to an energy eigenstate, write down a mathematical expression that relates the solutions of the one-dimensional time-dependent and time- independentSchro ̈dingerequations,Ψ(t,x)andψ(x).