Find those values of the separation constant e that allow

Question

Need to solve Part b and C only

Question In quantum mechanics, the position of a point particle in space is not certain - it is described by a probability distribution. The probability distribution of the position of the particle is |Ψ(x, t)|², where Ψ(x, t) is the wave function of the particle, which is a complex-valued function of position and time.

The one-dimensional, tune-dependent Schrodinger equation, describing the time evolution of the wave function Ψ(x, t) of a particle of mass m interacting with a potential V(x), is given by

ihΨ_t(x, t) = -h²/2mΨ_xx(x, t) + V(x)Ψ(x, t)

where h is a physical constant called the Planck constant. The potential V(x) describes the way the particle interacts with its environment, e.g. external forces or other particles.

(a) Use separation of variables to replace this partial differential equation with a pair of two ordinary differential equations. Denote the separation constant by E instead of -λ, since it turns out to be the total energy of the system.

(b) Let V(x) be the potential corresponding to an "infinite square well":

V(x) = { 0, 0 < x < 1,
         { ∞, x ≥ 1, or x ≤ 0.

Then Ψ(x,t) must be zero unless 0 < x < 1 and therefore Ψ(x, t) is the wave function of a particle trapped in a one-dimensional box, so this potential describes a particle surrounded by impermeable walls. This kind of potential could arise for example when modelling an electron orbiting an atomic nucleus. In this case, the Schrodinger equation reduces to

ihΨ_t(x, t) = -h²/2mΨ_xx(x, t),  0 < x < 1, t>0,

Ψ(0, t) = Ψ(1, t) = 0,                  t > 0.

Find those values of the separation constant E that allow for nonzero solutions. You should obtain an infinite number of values growing like n². This is an example of quantization of energy.

(c) With the potential V(x) as in part (b), suppose that initially the wave function is known to be

Ψ(x, 0) = 3/5sinΠx + 4/5sin3Πx.

Find Ψ(x, t) for all t > 0.