Question

Problem #1-basics \rightarrow photelectric effect, hydrogen atom, wave nature of electron (a) The work function of barium and tungsten are 2.5 eV and 4.2 eV respectively. Check whether these materials are useful in a photocell to detect visible light. The wavelength of visible light is in the range 4,000-7000 A. (10 pts) (b) How much energy is required to remove an electron from the $n = 8$ state of a hydrogen atom? What is the frequency of the radiation that just ionizes hydrogen atom (from the ground state)? (15 pts) (c) Estimate the energy of electrons needed to study electron diffraction of crystal structure if the interatomic spacing is $-2\AA$. (15 pts) Problem #2-uncertainty principle (15 pts) A bullet of mass 0.03 kg is moving with the velocity 500 m/s. The speed is measured to an accuracy of 0.02%. Calculate the uncertainty in position x (OK within a factor of 2). Comment on the result Problem #3-time-independent 1D Schrödinger Equation (20 pts) The energy eigenvalue and the corresponding eigenfunction for a particle of mass m in a one-dimensional potential V(x) are: $E = 0$ and $\psi (x) = \frac{A}{x^2 + a^2}$ Deduce the potential V(x). Problem #4-1D barrier (25 pts) Consider a particle moving from the left ($x = -\infty$) to the right in the potential (step) shown in the sketch. Use boundary matching conditions to express the reflected and transmitted probability amplitudes in terms of the incident amplitude.

          Problem #1-basics \rightarrow photelectric effect, hydrogen atom, wave nature of electron
(a) The work function of barium and tungsten are 2.5 eV and 4.2 eV respectively.
Check whether these materials are useful in a photocell to detect visible light. The
wavelength of visible light is in the range 4,000-7000 A. (10 pts)
(b) How much energy is required to remove an electron from the $n = 8$ state of a
hydrogen atom? What is the frequency of the radiation that just ionizes hydrogen atom
(from the ground state)? (15 pts)
(c) Estimate the energy of electrons needed to study electron diffraction of crystal
structure if the interatomic spacing is $-2\AA$. (15 pts)
Problem #2-uncertainty principle (15 pts)
A bullet of mass 0.03 kg is moving with the velocity 500 m/s. The speed is measured to
an accuracy of 0.02%. Calculate the uncertainty in position x (OK within a factor of 2).
Comment on the result
Problem #3-time-independent 1D Schrödinger Equation (20 pts)
The energy eigenvalue and the corresponding eigenfunction for a particle of mass m in a
one-dimensional potential V(x) are:
$E = 0$ and $\psi (x) = \frac{A}{x^2 + a^2}$
Deduce the potential V(x).
Problem #4-1D barrier (25 pts)
Consider a particle moving from the left ($x = -\infty$)
to the right in the potential (step) shown in the
sketch. Use boundary matching conditions to
express the reflected and transmitted probability
amplitudes in terms of the incident amplitude.

$Problem #1-basics →photelectric effect, hydrogen atom, wave nature of electron (a) The work function of barium and tungsten are 2.5 eV and 4.2 eV respectively. Check whether these materials are useful in a photocell to detect visible light. The wavelength of visible light is in the range 4,000-7000 A. (10 pts) (b) How much energy is required to remove an electron from the n = 8 state of a hydrogen atom? What is the frequency of the radiation that just ionizes hydrogen atom (from the ground state)? (15 pts) (c) Estimate the energy of electrons needed to study electron diffraction of crystal structure if the interatomic spacing is -2Å. (15 pts) Problem #2-uncertainty principle (15 pts) A bullet of mass 0.03 kg is moving with the velocity 500 m/s. The speed is measured to an accuracy of 0.02%. Calculate the uncertainty in position x (OK within a factor of 2). Comment on the result Problem #3-time-independent 1D Schrödinger Equation (20 pts) The energy eigenvalue and the corresponding eigenfunction for a particle of mass m in a one-dimensional potential V(x) are: E = 0 and ψ (x) = (A)/(x^2 + a^2) Deduce the potential V(x). Problem #4-1D barrier (25 pts) Consider a particle moving from the left (x = -∞) to the right in the potential (step) shown in the sketch. Use boundary matching conditions to express the reflected and transmitted probability amplitudes in terms of the incident amplitude.$

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