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Assignment 3: Bohr(-Sommerfeld) Model, de Broglie Hypothesis and
Uncertainty Principle
COURSE : P201-2023
JAN - APRIL 2023
Prof. VIJAY A. SINGH
physics.sutra@gmail.com
1. Einstein stated that the “Bohr Model appeared as a miracle .. and it appears as one
even today. This is the highest form of musicality in the realm of thought”. Discuss this
statement.
2. The Electron Microscope is based on the de Broglie hypothesis. Explore if and how it is
useful in Biology.
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3. Construct and label the phase-space plot of the classical one-dimensional harmonic oscillator of mass m, angular frequency ω and initial conditions x(0) = A and dx(0)/dt = 0.
Employ the Bohr-Sommerfeld quantization condition to obtain the energy levels of this
oscillator.
4. A stationary hydrogenic system of nuclear mass M = 6.63 × 10−27 kg and charge Z = 2
undergoes a transition of the shortest possible wavelength λ in the Balmer series. Estimate:
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(a) λ in Ao .
(b) The recoil speed of the atom.
5. Thermal Neutrons The mean kinetic energy per particle of an assembly of neutrons is 6.6
X 10−21 J.
(a) What is the de Broglie frequency of the matter wave associated with the neutron ?
(b) What is the de Broglie wavelength ?
(c) Would these neutrons exhibit diffraction effects when scattered off a crystal as in a
Davisson - Germer experiment ?
(d) What is the temperature of this assembly ?
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6. A monoenergetic beam of electrons is incident on a single slit of width 0.5 nm. A diffraction pattern is formed on the screen about 20 cm from the slit. The distance between
successive minima of the diffraction pattern is 2.1 cm. What is the energy of the incident
electron ?
7. A proton has a kinetic energy of 1 MeV. If the uncertainty in its momentum measurement
is 5 %. What is the minimum uncertainty in its position ?
8. Consider the electron in the Bohr model of the Hydrogen atom.
(a) What is the kinetic energy of the electron if its de Broglie wavelength is taken to be
of the order of the atomic diameter.
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(b) How does the answer in (a) compare with the ground state energy of the Hydrogen
atom ? Is it possible to “fix” this discrepancy ?
9. Semiconductor nanostructures : The drive towards the fabrication of smaller and even
smaller devices is currently at the submicron stage (0.1 - 1 µm devices). Research workers
are currently studying GaAs particles of size 10 nm (also called “artificial atoms”). Using
the uncertainty principle ∆p∆x ≃ h estimate the “upshift” in the electron energy due to
this extreme confinement. Would radiation from these particles be in the visible range ?
Note that the effective mass of the electron in GaAs is 0.06 the free electron mass and
the band gap of bulk GaAs is 1.5 eV at room temperature.
10. Using the unceratinty principle ∆p∆x ≥ ℏ/2. Obtain the ground state energy of a simple
harmonic oscillator of angular frequency ω.
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11. QUALITATIVE : Estimate the mass of a π - meson knowing that the effective distance
over which nuclear forces act is about 1.4 fermi.
12. Few discoveries in physics have attracted so much discussions inphilosophical, social and
other circles as the Uncertainty Principle. In this connection
(a) Name a few semincal discoveries in physics which have generated similar discussions.
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(b) What did Heisenberg himself think of this discouvery. [Hint: You may look at his
Nobel lecture for clues].
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