A three dimensional harmonic oscillator is in thermal equilibrium with a temperature reservoir at temperature T. The average total energy of the oscillator is :
Average K.E. in one dimension is
Average K.E. in 3 dimension is
The correct answer is:
Consider the particle in a three dimensional box problem for 2 boxes. One having equal edges of length a and the other of edges along x, y and z coordinates respectively.
The correct answer is: Ground state energy of the second box is greater
For the 3-dimensional second box:
First excited state, (nx, ny, nz) can be (1, 1, 2) or (1, 2, 1) or (2, 1, 1)
Now two energy levels have the same energy. Hence non degenerate
The correct answer is: The first excited state is non degenerate
What will be quantum number ‘n’ for earth? Take the earth sun distance to be 1.496 × 1011m
Energy levels for hydrogen atom are
Again substituting for the gravitational analog
here r is the earth sun distance
r = 1.496 × 1011m
n = 2.53 × 1074
The correct answer is: n ~ 1074
Which of the following is true about a quantum harmonic oscillator?
(1) A spectrum of evenly spaced energy states
(2) A potential energy function that is linear in the position coordinate
(3) A ground state characterized by zero kinetic energy.
(4) A non zero probability of finding the oscillator outside the classical turning points.
A quantum harmonic oscillator:
∴ Equally spaced
Potential Energy function
∴ quadratic in r.
For ground state,
A quantum Harmonic oscillator has non zero probability of being found beyond the classical turning points! This is one of the basic difference between a classical & quantum Harmonic oscillator
The correct answer is: 1 and 4 only
Which of the following functions could represent the radial wave function for an electron in an atom? (r is the distance of the electron from the nucleus; A, b are constants)
II. A sinbr
The wave function of an electron should satisfy the following properties
The only function (from the ones given) that satisfies these conditions is Ae–br
The correct answer is: I only
Suppose the earth made a transition to the next lower level (n – 1). How much energy (in Joules) would be released? What would be the wavelength of the emitted photon be?
Box I has each edge of length ‘a’ units. Box II has edges as a, 2a, a/2 along the x, y and z axis respectively. For what values of nx, ny, nz will the first excited state energy of box II be same as the first excited state energy of Box I
For nx - 1, ny = 2, nz = 1
(E1) box II = (E1) box I
The correct answer is: (nx, ny, nz) is (1, 2, 1)
What will be the Bohr radius of this system (work out the actual numerical value)
In the case of hydrogen atom,
for the gravitational analog.
= 2.34 × 10–138m
The correct answer is: r ~ 2 × 10–138m
Consider the Earth-Sun system as a gravitational analog to the hydrogen atom.
What is the potential energy function. (Let m be the mass of earth and M be the mass of the sun)
transforms hydrogen results to the gravitational analogs
(–)ve sign for attractive force.
The correct answer is: