Quantum Mechanics Answers

A Hermitian operator Aˆ has only three normalized eigenfunctions ψ1, ψ2, ψ3, with corresponding eigenvalues a1 = 1, a2 = 2, a3 = 3, respectively. For a particular state Φ of the system, there is a 50% chance that a measure of A produces a1 and equal chances for either a2 or a3. (a) Calculate hAi. (b) Express the normalized wave function Φ of the system in terms of the eigenfunctions of Aˆ.

If the real normalized functions f(x) and g(x) are not orthogonal, show that their sum f(x) +g(x) and their difference f(x)−g(x) are orthogonal. 

 Show that if the linear operators Aˆ and Bˆ do not commute, the operators (AˆBˆ + BˆAˆ) and i[A, ˆ Bˆ] are Hermitian. 

What is the difference between potential barrier and potential well ? [Mark-4]

7. Show that if the linear operators Aˆ and Bˆ do not commute, the operators

(AˆBˆ + BˆAˆ) and i[A, ˆ Bˆ] are Hermitian.

6. Show that xe−x

2

is an eigenfunction of the linear operator d

2

dx2 − 4x

2

.

What is the eigenvalue?

Calculate the probability of finding the particle in a one dimensional box of width

L, in state n within the intervals[0,L/4]

Using Bohr’s theory, calculate the energy required to remove the electron from the ground state of singly ionized helium, that is, helium with one electron already removed.

In the scattering of x-rays from a crystal of NaCl, it is assumed that the scattered x-rays undergo no change in wavelength. Show that this is a reasonable assumption by calculating the order of magnitude of the Compton wavelength for a sodium atom and a chlorine atom and comparing it with a typical x-ray wavelength of 0.10 nm.

How many photons per second are emitted by a 50-kW radio transmitter that operates at a frequency of 1200 kHz?