Irodov Solutions → Atomic and Nuclear Physics → Wave Properties of Particles. Schrodinger Equation 
6.49. Calculate the de Broglie wavelengths of an electron, proton, and uranium atom, all having the same kinetic energy 100 eV.
6.50. What amount of energy should be added to an electron to reduce its de Broglie wavelength from 100 to 50 pm?
6.51. A neutron with kinetic energy T = 25 eV strikes a stationary deuteron (heavy hydrogen nucleus). Find the de Broglie wavelengths of both particles in the frame of their centre of inertia.
6.53. Find the de Broglie wavelength of hydrogen molecules, which corresponds to their most probable velocity at room temperature.
6.54. Calculate the most probable de Broglie wavelength of hydrogen molecules being in thermodynamic equilibrium at room temperature.
6.55. Derive the expression for a de Broglie wavelength λ of a relativistic particle moving with kinetic energy T. At what values of T does the error in determining λ using the nonrelativistic formula not exceed 1% for an electron and a proton?
6.58. A parallel stream of monoenergetic electrons falls normally on a diaphragm with narrow square slit of width b = 1.0 μm. Find the velocity of the electrons if the width of the central diffraction maximum formed on a screen located at a distance l = 50 cm from the slit is equal to Δx = 0.36 mm.
6.59. A parallel stream of electrons accelerated by a potential difference V = 25 V falls normally on a diaphragm with two narrow slits separated by a distance d = 50 μm. Calculate the distance between neighbouring maxima of the diffraction pattern on a screen located at a distance l = 100 cm from the slits.
6.60. A narrow stream of monoenergetic electrons falls at an angle of incidence θ = 30° on the natural facet of an aluminium single crystal. The distance between the neighbouring crystal planes parallel to that facet is equal to d = 0.20 nm. The maximum mirror reflection is observed at a certain accelerating voltage V_{0}. Find V_{0} if the next maximum mirror reflection is known to be observed when the accelerating voltage is increased η = 2.25 times.
6.61. A narrow beam of monoenergetic electrons falls normally on the surface of a Ni single crystal. The reflection maximum of fourth order is observed in the direction forming an angle θ = 55° with the normal to the surface at the energy of the electrons equal to T = 180 eV. Calculate the corresponding value of the interplanar distance.
6.62. A narrow stream of electrons with kinetic energy T = 10 keV passes through a polycrystalline aluminium foil, forming a system of diffraction fringes on a screen. Calculate the interplanar distance corresponding to the reflection of third order from a certain system of crystal planes if it is responsible for a diffraction ring of diameter D = 3.20 cm. The distance between the foil and the screen is l = 10.0 cm.
6.63. A stream of electrons accelerated by a potential difference V falls on the surface of a metal whose inner potential is V_{i} = 15 V. Find:
6.65. Describe the Bohr quantum conditions in terms of the wave theory: demonstrate that an electron in a hydrogen atom can move only along those round orbits which accommodate a whole number of de Broglie waves.
6.66. Estimate the minimum errors in determining the velocity of an electron, a proton, and a ball of mass of 1 mg if the coordinates of the particles and of the centre of the ball are known with uncertainly 1 μm.
6.68. Show that for the particle whose coordinate uncertainty is Δx = λ/2π, where λ is its de Broglie wavelength, the velocity uncertainty is of the same order of magnitude as the particle's velocity itself.
6.70. Employing the uncertainty principle, estimate the minimum kinetic energy of an electron confined within a region whose size is l = 0.20 nm.
6.71. An electron with kinetic energy T ≈ 4 eV is confined within a region whose linear dimension is l = 1 μm. Using the uncertainty principle, evaluate the relative uncertainty of its velocity.
6.73. A particle of mass m moves in a unidimensional potential field U = kx^{2}/2 (harmonic oscillator). Using the uncertainty principle, evaluate the minimum permitted energy of the particle in that field.
6.82. A particle is located in a twodimensional square potential well with absolutely impenetrable walls (0 < x < a, 0 < y < b). Find the probability of the particle with the lowest energy to be located within a region 0 < x < a/3.
6.83. A particle of mass m is located in a threedimensional cubic potential well with absolutely impenetrable walls. The side of the cube is equal to a. Find:
6.84. Using the Schrodinger equation, demonstrate that at the point where the potential energy U(x) of a particle has a finite discontinuity, the wave function remains smooth, i.e. its first derivative with respect to the coordinate is continuous.
6.85. A particle of mass m is located in a unidimensional potential field U(x) whose shape is shown in Fig. 6.2, where U(0) = ∞. Find:
6.90. The wave function of a particle of mass m in a unidimensional potential field U(x) = kx^{2}/2 has in the ground state the form ψ(x) = Ae^{αx2}, where A is a normalization factor and α is a positive constant. Making use of the Schrodinger equation, find the constant α and the energy E of the particle in this state.
6.92. The wave function of an electron of a hydrogen atom in the ground state takes the form ψ(r) = Ae^{r/r1}, where A is a certain constant, r_{1} is the first Bohr radius. Find:
6.93. Find the mean electrostatic potential produced by an electron in the centre of a hydrogen atom if the electron is in the ground state for which the wave function is ψ(r) = Ae^{r/r1}, where A is a certain constant, r_{1} is the first Bohr radius.
6.95. Employing Eq. (6.2e), find the probability D of an electron with energy E tunnelling through a potential barrier of width l and height U_{0} provided the barrier is shaped as shown:
6.96. Using Eq. (6.2e), find the probability D of a particle of mass m and energy E tunnelling through the potential barrier shown in Fig. 6.6, where U(x) = U_{0}(1  x^{2}/l^{2}).
