# Exercises (Problems)

- Page ID
- 49051

## 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

- 100-W yellow light bulb (\(λ= 550\;nm\))
- 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.

### Q4.4.3

## 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:

- a .8g bullet with velocity 340ms
^{-1}. - a 10
^{-5}g particle with velocity 10^{-5}ms^{-1}. - a 10
^{-8}g particle with velocity 10^{-8}ms^{-1}. - an electron moving with velocity 4.8*10
^{6}ms^{-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.

- What is the momentum of a galloping horse? What is its wavelength?
- 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?
- 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
- \( \hat{B} = \dfrac{d^2}{dx^2}\)

- \(3e^{-cx^3}\)
- \(\cos(4ax^2)\)
- \(e^{i^2kx^3}\)

## 4.8: Particle in a One-Dimensional Box

### Q4.8.1

### Q4.8.2

### Q4.8.3

- 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.
- 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^{ o}C, 0 ^{o}C, and 100^{ o}C.

## 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 *m _{l}* 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?