Solved Problems

1. (a) What is the size of an atom?

(b) What is the velocity of an electron in the first Bohr orbit of an Hydrogen atom?

Solution

a) You can estimate size of an atom as

Or you can calculate Bohr radius that gives size of an atom:

quantization of angular momentum

Force between proton and electron. Eliminating velocity between equations we

obtain:

For n=1; substituting:

b) Quantization of angular momentum yields:

Compare to the speed of light:

2. Calculate frequency and energy of the photons, in eV, having a wavelength:

a) Radio wave 900 meter

b) Microwave 3.0 cm

c) Visible light range 500 nm

d) x-ray range 5.0 nm

e) Gamma ray 1.0 pm

Solution

Energy of the photon is given by ; c=3

a) OR

Think!! Which of the wave is more energetic? What is the energy of FM radio

signals? Why are Gamma rays dangerous??

You can solve the remaining part of the problem.

3. Radiation has been detected from space that is characteristic of an ideal radiator at

(Relic from “Big-Bang”). For this temperature at what wavelength does

the Planck distribution peak? In what part of the electromagnetic spectrum is this

wavelength?

Solution

Wien’s displacement law state maximum value of the wavelength of radiation from an

object:

This wavelength is in the microwave part of the spectrum.

4. A clean nickel surface is exposed to light of wavelength of 235nm. What is the

maximum speed of the photoelectrons emitted from this surface. The work function

for nickel is .

Solution

Using the relation

5. The threshold of dark-adapted (scotopic) vision is at acentral

wavelength of 500 nm. If light with this intensity and wavelength enters the eye when

the pupil is open to its maximum diameter of 8.5 mm, how many photons per second

enter the

eye?

Solution:

Scotopic is means; energy of the radiation per second per meter square.

We can calculate area of pupil and then power of the light entering the eye.

Power of the light entering the eye:

This is the energy of the light entering to the ey eper second. We calculate energy of each

photon:

Number of photon per second:

6. A beam of 0.68-nm photons undergoes Compton scattering from free electrons. What

are the energy and momentum of the photons that emerge at a 45° angle with respect

to the incident beam?

Solution :

Using the Compton shift formula, then we can write

Therefore the energy of the photon is

Momentum of the photon is:

7. Calculate the de Broglie wavelength for an electron that has kinetic energy (a) 50.0 eV

and (b) 50.0 keV (ignore relativistic effects).

Solution :

The de Broglie wavelength is , where p is the linear momentum. If relativistic effects are

ignored, then one can easily relate the momentum to so

(a) If and the particle is an electron,then

(b) For 50.0 keV electrons,

8. A hydrogen atom is in its first excited state (n = 2). Using the Bohr theory of the atom,

calculate (a) the radius of the orbit, (b) the linear momentum of the electron, (c) the

angular

momentum of the electron, (d) the kinetic energy, (e) the potential energy, and (f) the total

energy.

Solution :

(a) See problem 1.

(b) And (c)

(c) Firstly let us calculate angular acceleration

Then we can calculate p:

(d) Kinetic Energy

(e) Potential Energy:

(f) Total Energy:

9. Write down time dependent Schrödinger equation.

Answer. Time dependent Schrödinger equation can be written as:

10. Show that is the solution of the Schrödinger equation for

. (

Solution: For we can write

Substitute given wave function into the Schrödinger equation

Evaluating derivatives:

is satisfied.

11. What is the properties of wave function . (Psi)

Answer: Read the lecture notes

12. Write down momentum and position operator.

Answer: Observables of classical physics can be represented by an operator. The momentum

and position operator are .

13. Find the commutation relation between the position and momentum operators.

Focus your attention!! is a differential operator. Then Wrong. Evaluation as

follows:

Then remove f and: ; substitute the result into the above relation:

14. Find the eigenfunctions of the operators .

Solution.

Eigenvalue equation can be written as where is operator is its eigenvalue and

is eigenfunction of .

Then

Integrating both sides yields

Similarly

Solution of the differential equation yields

15. Find the eigenfunctions of the operators have the

same eigenfunction.

Solution. Let us find eigenfunction of A

Solving the equation, we obtain

You can calculate eigenfunction of .

16. Calculate commutator of A and

Solution