Homework 2

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Q2.1

Calculate the de Broglie wavelength and the kinetic energy of each of the following:

1. A 100 kg man jogging on a treadmill with a velocity of 1 m/s
2. A 10 g silver bullet fired at a velocity of 1000 m/s
3. The moon as it orbits the Earth
4. A positron with a velocity of \(7 \times 10^6\) cm/s
5. A chicken crossing the road with a velocity of 1 cm/s. Assume the chicken has a mass of 1 kg.
6. An electron in the n = 1 Bohr orbital which has a velocity of \( 2.19 \times 10^6\) m/s

Q2.2

Estimate the de Broglie wavelength of electrons that have been accelerated from rest through a potential difference (V) of 100 kV. (Hint: kinetic energy is eV).

Q2.3

A thermalized electron is an electron with the kinetic energy given by \(k_BT\) with \(k_B\) as the Boltzmann constant. Calculate the de Broglie wavelength of a thermalized electron at 295 K.

Q2.4

Estimate the theoretical resolution of an electron microscope that produces electrons from a tungsten filament with an energy of 200 keV.

Hint: The resolution of a microscope is the shortest distance between two points on a sample that can still be distinguished by the observer or camera system as separate entities. This is best estimated as half the wavelength of the incident particles for this discussion although more complex formulations exist.

Q2.5

Calculate the longest and shortest wavelengths of the highest and lowest energy of the Paschen emission line of hydrogen.

1. What is the angular momentum in the lowest energy state of the Bohr hydrogen atom?
2. What is the velocity of the electron in the lowest energy state?

Q2.6

1. What is the minimum uncertainly in the velocity of an arrow (mass = 0.5 kg) that lands within \(1.0 \times 10^8\;\) m of a target?
2. What is the maximum uncertainty in the position of a rock (mass = 1 kg) with a speed somewhere between 632.00001 m/s and 632.00000 m/s?

Q2.7

A quantum mechanical turtle crosses the road at a velocity of \(1 \times 10^{16}\) nm/sec, with an uncertainty of \(10^{-283}\) mm/sec. Calculate the uncertainty in location of the turtle. (Assume the turtle has a mass of 1 kg.)

Q2.8

Show that the functions \(e^{i(k x + ωt)}\) and \(\cos(k\,x - \omega\, t)\) also satisfy the classical wave equation. Note that \(i\) is a constant equal to \(\sqrt {-1}\).

Q2.9

For the linear homogeneous differential equation

\[ \dfrac{d^2u(t)}{dt^2} + \omega^2 u(t) = 0\ \label{2.0}\]

the general solution is

\[ u(t) = A \cos(\omega t) + B \sin(\omega t) \label{2.1}\]

which can also be rewritten to

\[ u(t) = C \cos(\omega t + \phi) \label{2.2}\]

or

\[ u(t) = C \sin(\omega t + \psi) \label{2.3}\]

• Show that Equations \(\ref{2.1}\) through \(\ref{2.3}\) are equivalent
• Demonstrate that each is the solution to the original differential equation (Equation \(\ref{2.0}\)).
• Derive the relationships between \(C\) and \(\phi\) in terms of \(A\) and \(B\).
• Derive the relationships between \(C\) and \(\psi\) in terms of \(A\) and \(B\).

Hint: You may need to use the following trigonometric identities

\[\sin( \alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\]

\[\cos( \alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\]

Q2.10

The spatial part of the wavefunction of a particle constrained between 0 and L and can be described as:

\[ \dfrac{\partial^2 u(x)}{\partial x^2} + \left( \dfrac{8\pi^2m E}{h^2} \right) u(x)= 0 \label{2.4}\]

with the boundary condition:

\[u(0)= u(L) = 0 \label{2.5}\]

For this system, \(E\) is the energy of the particle and u(x) describes the spatial part of its wave nature. Solve Equation \(\ref{2.4}\) subject to the boundary constraints in Equation \(\ref{2.5}\). What values of energy are possible? Is the system quantized and why or why not?

Q2.11

The total energy is the sum of the Kinetic and Potential energies (E = K.E. + V). The exponential form of the solution to the wave equation is:

\[ \Psi = a e^{2\pi i x / \lambda} e^{-2\pi i \nu t} \label{2.6} \]

And the de Broglie relation is:

\[\lambda = \dfrac{h}{mv} = \dfrac{h}{p} \label{2.7} \]

1. Rewrite the generalized 1D time-dependent wave equation in terms of energy (instead of wavelength).
2. Assume that an electron with this wavefunction moves in a finite potential that is greater than \(E\). What is the behavior of the wavefunction at this potential?