#2 Homework
- Page ID
- 120314
Name: ______________________________
Section: _____________________________
Student ID#:__________________________
Print this sheet out, but write the solutions to these problems on a separate sheet of paper. You will need the space to write the problems clearly.
Q1
Estimate the de Broglie wavelength of electrons that have been accelerated from rest through a potential difference (V) of 30 kV. (Hint: kinetic energy is eV).
Q2
Calculate the de Broglie wavelength of a room temperature thermalized neutron (i.e., having a kinetic energy of \(k_BT\) with \(k_B\) as the Boltzmann constant).
Q3
Calculate the wavelength of the highest energy Balmer emission line of hydrogen.
- What is the angular momentum in the lowest energy state of the Bohr hydrogen atom?
- What velocity of the electron in the lowest energy state
Q4
What is the minimum uncertainly in the speed of a 1 kg ball that is known to to be within \(1.0 \times 10^6\;m\) on a bat? What is the maximum uncertainy in the position of a 10 g carrot with a speed somewhere between 250.00001 m/s and 250.00000 m/s?
Q5
Calculate the uncertainty in location of a 100 g (slow) chicken crossing a road at a velocity of 1 mm/sec, with an uncertainty of \(10^{-4} mm/sec\).
Q6
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 \(\sqrt {-1}\).
Q7
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
\[\sin( \alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta\]
\[\cos( \alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\]
Q8
Is a particle is constraints to the between 0 and L and can be describe with the following wave equation (spatial part only)
\[ \dfrac{\partial^2 u(x)}{\partial x^2} + \left( \dfrac{8\pi^2m E}{h^2} \right) u(x)= 0 \label{3.1}\]
with the boundary condition used in the string example
\[u(0)= u(L) = 0 \label{3.2}\]
For this system, \(E\) is the energy of the particle and u(x) describes the spatial part of its wave nature. Solve Equation \(\ref{3.1}\) subject to the boundary constraints in Equation \(\ref{3.2}\). What values of energy are possible? Is the system quantized and why or why not?