Phys 301show that the wavefunction is normalizedcalculate

Question

1. For any eigenfunction ψ_n;.of the infinite square well,

(a)Show that =L/2.

(b)Show that 2>=(L²/3)-(L2/2(nΠ)^2),

where L is the size of the well.

2. Consider a system whose wavefunction at t= 0 is
Ψ(x,0) = (1 /5)Φ₁(x) +(3/√ 20)Φ₂ (x)+(√(7/20))Φ₃(x),

where Φ_n(x) is the eigenfunction of the n^th state of an infinite square well potential of width I with the energy eigenvalues

E_n = Π²? ²n²/(2ma²)^.

(a) Show that the wavefunction is normalized.

(b) Calculate the average energy of this system.

(c) Find the state ψ(x, t) at any later time Oand evaluate the average value of the
energy. Compare the result with the value obtained in (a). Does it depend upon time and why ?

(d) What are the frequencies found in oscillations of the probability density and what is the periodicity of the oscillations, i.e., after what time interval does the probability density return to its initial value?
3. The nuclear potential that binds protons and neutrons in the nucleus of an atom is often approximated by a square well. Imagine a proton confined in an infinite square well of length 10^-5 nm, a typical nuclear diameter. Calculate the wavelength and energy associated with the photon that is emitted when a proton makes a transition from the first excited state (n = 2) to the ground state (n= 1). In what region of the electromagnetic spectrum does this wavelength belong ?