# 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. Free solution >> 6.50. What amount of energy should be added to an electron to reduce its de Broglie wavelength from 100 to 50 pm? Free solution >> 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. Free solution >> 6.53. Find the de Broglie wavelength of hydrogen molecules, which corresponds to their most probable velocity at room temperature. Free solution >> 6.54. Calculate the most probable de Broglie wavelength of hydrogen molecules being in thermodynamic equilibrium at room temperature. Free solution >> 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 non-relativistic formula not exceed 1% for an electron and a proton? Free solution >> 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. Free solution >> 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. Free solution >> 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 V0. Find V0 if the next maximum mirror reflection is known to be observed when the accelerating voltage is increased η = 2.25 times. Free solution >> 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. Free solution >> 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. Free solution >> 6.63. A stream of electrons accelerated by a potential difference V falls on the surface of a metal whose inner potential is Vi = 15 V. Find: (a) the refractive index of the metal for the electrons accelerated by a potential difference V = 150 V; (b) the values of the ratio V/Vi at which the refractive index differs from unity by not more than η = 1.0%. Free solution >> 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. Free solution >> 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. Free solution >> 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. Free solution >> 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. Free solution >> 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. Free solution >> 6.73. A particle of mass m moves in a unidimensional potential field U = kx2/2 (harmonic oscillator). Using the uncertainty principle, evaluate the minimum permitted energy of the particle in that field. Free solution >> 6.82. A particle is located in a two-dimensional 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. Free solution >> 6.83. A particle of mass m is located in a three-dimensional cubic potential well with absolutely impenetrable walls. The side of the cube is equal to a. Find: (a) the proper values of energy of the particle; (b) the energy difference between the third and fourth levels; (c) the energy of the sixth level and the number of states (the degree of degeneracy) corresponding to that level. Free solution >> 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. Free solution >> 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: (a) the equation defining the possible values of energy of the particle in the region E < U0; reduce that equation to the form sin kl = ±kl sqrt(ħ2/2ml2U0), where k = sqrt(2mE)/ħ. Solving this equation by graphical means, demonstrate that the possible values of energy of the particle form a discontinuous spectrum; (b) the minimum value of the quantity l2U0 at which the first energy level appears in the region E < U0. At what minimum value of l2U0 does the nth level appear? Free solution >> 6.90. The wave function of a particle of mass m in a unidimensional potential field U(x) = kx2/2 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. Free solution >> 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, r1 is the first Bohr radius. Find: (a) the most probable distance between the electron and the nucleus; (b) the mean value of modulus of the Coulomb force acting on the electron; (c) the mean value of the potential energy of the electron in the field of the nucleus. Free solution >> 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, r1 is the first Bohr radius. Free solution >> 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 U0 provided the barrier is shaped as shown: (a) in Fig. 6.4; (b) in Fig. 6.5. Free solution >> 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) = U0(1 - x2/l2). Free solution >>