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Section 11: Modern Physics

1. De Broglie Wavelength

  • Calculate the de Broglie wavelength of an electron with a kinetic energy of 150 eV.

  • Calculate the de Broglie wavelength of a 50-gram golf ball traveling at a speed of 60 m/s.

  • Why are quantum effects not noticeable for macroscopic objects?

2. X-Ray Production

What is the minimum potential difference that must be applied across an X-ray tube to produce X-rays with a wavelength of \(0.01\) nm? (Hint: use the formula \(E = hf\) and \(E = qV\)).

3. Photoelectric Effect

The work function of potassium is 2.3 eV. What is the maximum kinetic energy of electrons ejected from potassium when it is illuminated with ultraviolet light of wavelength 350 nm?

4. Radius and Angular Momentum

Using the Bohr model, determine the orbital radius and angular momentum of an electron in the hydrogen atom for the state \(n=2\). Express the radius in meters and in units of the Bohr radius

\[ a_0 = \frac{4 \pi \varepsilon_0 \hbar^2}{m_e e^2}, \]

and express angular momentum in units of \(\hbar\).

5. Heisenberg Principle

Using the Heisenberg Uncertainty Principle (\(\Delta x \Delta p \ge \hbar/2\)), what is the minimum uncertainty in the velocity of an electron that is confined within a region of space 0.1 nm wide (which is approximately the size of an atom)?

6. Quantum Numbers

For the \(n=3\) energy level in a hydrogen atom, what are the possible values for the quantum numbers \(l\) (orbital) and \(m_l\) (magnetic)? How many distinct electron states exist for \(n=3\)?

7. Hydrogen Emission Spectra

Calculate the energy of the photon emitted by a hydrogen atom when an electron transitions from the \(n=4\) state to the \(n=2\) state. What is the wavelength of this photon? What is the color of this light in the visible spectrum? (Hint: Use the Rydberg formula for hydrogen levels: \(E_n = -13.6\,\text{eV}/n^2\)).

8. Quantum Well Spectra

An electron is in a 1D infinite potential well of width \(L = 0.5\) nm. What is the wavelength of photon emitted when the electron transitions from the \(n=4\) state to the \(n=2\) state?

9. Pair Annihilation

An electron and a positron, each with a rest mass of \(0.511 \text{MeV/c}^2\), annihilate each other, producing two photons of equal energy. What is the energy (in MeV) and wavelength of each photon?

10. Radioactive Half-Life

The half-life of Cobalt-60 is 5.27 years. If a sample initially contains 100 grams of Cobalt-60, how much will remain after approximately 21 years?

11. Alpha Decay

Give a specific, balanced nuclear equation for an alpha decay process, starting with Uranium-238 (\({}^{238}_{92}\text{U}\)).

12. Beta Decay

Give a specific, balanced nuclear equation for a beta-minus decay process, starting with Carbon-14 (\({}^{14}_{6}\text{C}\)).

13. Wavefunction Probability

For a particle in a 1D box of length L, the wavefunction for the ground state is \(\Psi(x) = \sqrt{2/L} \sin(\pi x/L)\). Calculate the probability of finding the particle in the region \(0 \le x \le L/4\). Then calculate the probability of finding the particle in the region \(L/4 \le x \le L/2\). Which region is more likely to contain the particle?

14. Hadamard Gate and the “Schrödinger’s Cat” State

The basis states of a qubit can be written as column vectors:

\[ |0\rangle = \begin{pmatrix} 1\\ 0 \end{pmatrix}, \qquad |1\rangle = \begin{pmatrix} 0\\ 1 \end{pmatrix}. \]

The Hadamard gate \(H\), represented by the following \(2\times 2\) matrix, acts on the state \(|0\rangle\):

\[ H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}. \]

Show that after applying the Hadamard gate, the final state has the form

\[ |\psi\rangle = \frac{1}{\sqrt{2}}|0\rangle + \frac{1}{\sqrt{2}}|1\rangle. \]

Then explain why this state is a superposition of the two basis states and why it is sometimes described, in an illustrative sense, as a “Schrödinger’s cat” state.