# Exercises (Problems)

## 4.1: The Wave Theory of Light

### Q4.1.1

What is the wavelength associated with a photon of a light with the energy is $$3.6 \times 10^{-19}\;J$$?

### Q4.1.2

Calculate the energy of a photon of a light with the frequency is $$6.5 \times 10^{-14}\; s^{-1}$$ ?

### Q4.1.3

Calculate the energy per photon and the number of photons emitted per minute from

1. 100-W yellow light bulb ($$λ= 550\;nm$$)
2. 1-kW microwave source ($$λ = 1\;cm$$)

## 4.3: The Photoelectric Effect

### Q4.3.1

If the wavelength of a x-ray photon in a x-ray photoelectron spectroscopy (XPS) instrument is 1.25 nm. Calculate the velocity of the electrons emitted from molecules in which the following work functions (e.g., binding energies): 25, 125, 425 eV.

## 4.4: Bohr's Theory of the Hydrogen Emission Spectrum

### Q4.4.2

Calculate the wavenumber of the wavelength of the light emitted from the $$n=8$$ to $$n=6$$ transition.

## 4.5: de Broglie's Postulate

### Q4.5.1

Calculate the wavelength associated with a 42 g baseball with speed of 80 m/s.

### Q4.5.2

Calculate the de Broglie wavelengths of the following:

1. a .8g bullet with velocity 340ms-1.
2. a 10-5g particle with velocity 10-5ms-1.
3. a 10-8g particle with velocity 10-8ms-1.
4. an electron moving with velocity 4.8*106ms-1.

### Q4.5.3

What is the de Broglie wavelength of a thermal neutron at 350 K? A thermal neutron has the kinetic energy equal to the average kinetic energy of a thermalized monotonic gas (i.e., described by the Maxwell-Boltzmann distribution).

## 4.6: The Heisenberg Uncertainty Principle

### Q4.6.1

If the uncertainty of measuring the position of an electron is 2.0 Å, what is the uncertainty of simultaneously measuring its velocity? Hint: What formula deals with uncertainty of measurements?

### Q4.6.2

A typical mass for a horse is 510 kg, and a typical galloping speed is 22 kilometers per hour. Use these values to answer the following questions.

1. What is the momentum of a galloping horse? What is its wavelength?
2. If a galloping horse's velocity and position are simultaneously measured, and the velocity is measured to within ± 1.0%, what is the uncertainty of its position?
3. Suppose Planck's constant was actually 0.01 J s. How would that change your answers to (a) and (b)? Which values would be unchanged?

Hints:

• de Broglie's postulate deals with the wave-like properties of particles.
• Heisenberg's uncertainty principle deals with uncertainty of simultaneous measurements.

### Q4.6.3

Consider a balloon with a diameter of $$2.5 \times 10^{-5}\; m$$. What is the uncertainty of the velocity of an oxygen molecule that is trapped inside.

## 4.7: The Schrödinger Wave Equation

### Q4.7.1

What are the results of operating on the following functions with following two operators:

• $$\hat{A} = \dfrac{d}{dx}$$ and
• $$\hat{B} = \dfrac{d^2}{dx^2}$$
1. $$3e^{-cx^3}$$
2. $$\cos(4ax^2)$$
3. $$e^{i^2kx^3}$$

## 4.8: Particle in a One-Dimensional Box

### Q4.8.3

1. Calculate the energy levels for n = 1, 3, and 5 for an electron in a potential well of width 0.50 nm with infinite barriers on either side.
2. If an electron makes a transition from n = 3 to n = 1, what will be the wavelength of the emitted radiation?

### Q4.8.4

For a helium atom in a one-dimensional box, calculate the quantum number for the wavefunctions with the energies equal to the average kinetic energy of a thermalized monotonic gas (e.g, 3/2 kT) for a box 1 nm long at -100 oC, 0 oC, and 100 oC.

## 4.10: The Schrödinger Wave Equation for the Hydrogen Atom

### Q4.10.1

Indicate the number of subshells, the number of orbitals in each subshell, and the values of l and ml for the orbitals in the n = 4 shell of an atom.

### Q4.10.2

Identify the subshell in which electrons with the following quantum numbers are found: (a) n = 3, l = 1; (b) n = 5, l = 3; (c) n = 2, l = 0.

### Q4.10.3

Calculate the maximum number of electrons that can occupy a shell with (a) n = 2, (b) n = 5, and (c) n as a variable. Note you are only looking at the orbitals with the specified n value, not those at lower energies.

### Q4.10.4

How many orbitals have l = 2 and n = 3?