Homework 2

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Q2.1

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

1. A 20 kg (really heavy) frisbee with a velocity of 2 m/s
2. A 2 g pineapple thrown at a velocity of 20,000 m/s
3. The moon as it orbits the Earth
4. An electron with a velocity of \(1 \times 10^6\) cm/s
5. An electron in the n = 1 Bohr orbital which has a velocity of \( 2.19 \times 10^6\) m/s

Q2.2

The standard accelerating voltage for a scanning electron microscope is 20 keV. Find the de Broglie wavelength of electrons that have been accelerated from rest through that potential difference (V) of 20 keV. (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 298 K.

Q2.4

1. What is the minimum uncertainly in the velocity of a projectile (mass = 1 kg) that lands within \(10.0 \times 10^6\;\) m of a target?
2. What is the maximum uncertainty in the position of a rabbit (mass = 2 kg) with a speed somewhere between 200.000001 m/s and 200.00000 m/s?

 

Q2.5

A quantum mechanical chicken crosses the sidewalk at a velocity of \(1 \times 10^{32}\) nm/sec, with an uncertainty of \(10^{-295}\) mm/sec. Calculate the uncertainty in location of the chicken. (Assume the chicken has a mass of 1 kg.)

Q2.6

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 \(\sqrt {-1}\).

Q2.7

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

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

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

Hint: You may need to use the following trigonometric identities

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

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

Q2.8

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.

1. Solve Equation \(\ref{2.4}\) subject to the boundary constraints in Equation \(\ref{2.5}\).
2. What values of energy are possible?
3. Is the system quantized and why or why not?

 

Q2.9

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?