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#2 Homework

  • Page ID
    120314
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    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.

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


    #2 Homework is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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