Download Introduction to Quantum Mechanics: Homework #1 (Due by Sep

Introduction to Quantum Mechanics
Homework #1 (Due on March 24, 2014)
1. A particle of mass m= 1 kg is moving on a plane in which the potential is
given by V(x, y) = 4x2 + y2, where x and y are in m. The particle is initially at
the point (1,√3) and v(0) = 0.
1) Express the force F(x, y).
2) Obtain the trajectory r = r (t) of the particle and draw it on the x-y plane.
Further show that the trajectory is an ellipse.
3) What are the v, kinetic energy Ek (J), and V(J) of the particle at t = 5 s?
4) Find the point(s) where the speed of the particle is maximum.
2. 1) Calculate ∫0∞ x2n exp (-x2) dx for odd and even integers n, respectively.
2) Calculate Σ0∞ n exp (-nx)
3. The energy of a classical 1-D harmonic oscillator is given by E = T + V = ½
mv2 + ½ kx2, where v and x can take any real value. Applying Boltzmann’s
energy distribution law, show that the average energy of an 1-D HO at
temperature T is given by
<E> = <T> + <V> = ½ kT + ½ kT = kT
4. Applying the Planck hyphothesis E = n hν (n = 0,1,2,3 …) for a harmonic
oscillator with a characteristic frequency ν and Boltzmann’s energy distribution
law, calculate <E> under thermal equilibrium at temperature T.
5. 1) Using the Planck distribution, show that Tλmax ~ hc /5k .
2) Calculate the λmax for a blackbody at 2000 K.
6. Consider a grey body whose emissivity ε (λ) = 0.6 at all λ. How much energy
is emitted from the grey body at T =1500 K per unit surface area in 1 sec?
7. Based on the Einstein’s model for the heat capacity Cv of a solid, show that
1) Cv→0 as T→0 and 2) Cv = 3R at high T.
8. Suppose one performs photoelectron spectroscopy of Ni (work function of φ
= 5.0 eV) sample using a 10 mW ultraviolet light source emitting at λ= 58.4
nm. Assume that all the photons emitted by the source irradiate the Ni
surface. 1 eV =1.6x10-19 J; it is equal to the kinetic energy gained by an
electron accelerated by an electrical potential of 1 V.
1) Calculate the number of photons incident on the sample surface per
second.
2) Calculate the maximum kinetic energy of the photoelectrons emitted from
the sample.
3) Calculate the shortest De Broglie wavelength of the emitted electrons in Å
unit.
4) If all the photoelecrtrons are collected, what is the current in ampere (A)?
5) Calculate the maximum wavelength of a UV light with which you can
observe the photoelectric effect from the Ni sample.
9. Based on the Bohr model of a H atom,
1) Show that the Bohr radius (the radius of the smallest orbital) a = 0.529 Ǻ.
2) Show that the ionization energy of the electron in ground (lowest energy)
state H atom is 13.6 eV.
3) What is the ground state energy of He+ ion? (Hint: Z = 2)
3) Calculate the λ (Å) of the light emitted by H atom in the electronic transition
n = 2 → n = 1, where n is the quantum number.
4) Calculate the Rydberg constant R.
10. Show that
1) A linearly polarized light Ψ(x, t) = Ψ0 sin (kx –ωt) traveling in the +x direction
satisfies the (master) 1D wave equation ∂2 Ψ/∂x2 = (1/v2) ∂2 Ψ/∂t2 when v = c.
2) A spherical light wave Ψ(r, t) = Ψ0 exp (k . r – ωt)/r satisfies the 3D wave
equation ∇2 Ψ = (1/c2) ∂2 Ψ/∂t2.
11. Find an eigenfunction and the corresponding eigenvalue for each of the
following operators; 1) d/dx, 2) d2/dx2, 3) d2/dx2 + a, 4)▽2 + a, where a is a
constant.
12. Consider two different free particle (mass of m) states represented by Ψ1 = 3
exp (ikx) and Ψ2 = A exp (-2ikx).
1) Show that Ψ1 and Ψ2 are the eigenfunctions of the momentum operator px.
What are the corresponding eigenvalues of px for Ψ1 and Ψ2, respectively.
2) Express the energy eigenvalues of the Ψ1 and Ψ2 states, respectively.
3) Consider a mixed state Ψ = Ψ1 + Ψ2. Show that Ψ is not an eigenfunctions
of the momentum operator px. In this case, the particle does not have a
definite momentum. If one measures its momentum px, what would be the
average (expectation) value <px>?
4) If one measures the momentum px, what values would be resulted and
what is the probability to observe each value?
5) Show that Ψ is not an eigenfunction of the operator H.
6) If one measures the energy px, what values would be resulted and what is
the probability to observe each value?
13. The wavefunction for a particle in 1D is given by ψ(x) = N exp (-x2/4).
1) Normalize ψ(x).
2) Show that this is not an eigenfunction of the position and momentum
operators.
3) Calculate the probability to find the particle in -1≦ x ≦1.
4) Calculate the expectation values of <x> and <px>.
5) Show that the uncertainty principle δx. δpx ≥ h/4π holds for this particle.