– The de Broglie wavelength of a particle with momentum p:
\[
2=\frac{2 n t}{p} \text {. }
\]
– Uscertainty prisciple:
\[
\Delta x \cdot \Delta p_{x}>\mathbf{A} \text {. }
\]
– Schrblinger time-dependent asd time-independent equations:
\[
\begin{array}{l}
\text { in } \frac{\partial \psi}{\partial t}=-\frac{b^{2}}{2 m}
abla \cdot \psi+U \Psi, \\

abla v+\frac{2 m}{n d}(E-U) v=0,
\end{array}
\]
where $\mathbf{T}$ is the lotal save function, $\mathbf{\phi}$ is its coordinate gart, $\mathbf{v}$ is the Laplace operator, $E$ and $U$ are the total and potential energies of the particle. In spiericaf coondinates:
– Ceelficient of tranaparency of a petential harrier $V(x):$
\[
D \approx \exp \left[-\frac{2}{n} \int_{i}^{x} \sqrt{25(V-E)} d x\right] .
\]
where $s_{1}$ and $z_{2}$ are the coordinates of the points bet veen which $V>E$.
6.49. Calculate the de Broglie wavelengths of an electron, proton, and uranium atom, all having the same kinetic energy $100 \mathrm{eV}$.
6.50. What amount of energy should be added to an electron to reduce its de Broglie wavelength from 100 to $50 \mathrm{pm}$ ?
6.51. A neutron with kinetic energy $T=25 \mathrm{eV}$ strikes a stationary deuteron (heavy hydrogen nucleus). Find the de Broglie wavelengths of both particles in the frame of their centre of inertia.
6.52. Two identical non-relativistie particles move at right angles to each other, possessing de Broglie wavelengths $\lambda_{1}$ and $\lambda_{1}$; Find the de Broglie wavelength of each particle in the frame of their centre of inertia.
6.53. Find the de Broglie wavelength of hydregen 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 thermedynamic equilibrium at room temperature.
6.55. Derive the expression for a de Breglie wavelength $\lambda$ of a relativistic particle moving with kinetic energy $T$. At what values of $T$ does the error in determining $\lambda$ using the non-relativistic formula not exceed $1 \%$ for an electron and a proton?
6.56. At what value of kinetic energy is the de Broglie wavelength of an electron equal to its Compton wavelength?
6.57. Find the de Broglie wavelength of relativistic electrons reaching the anticathode of an $\mathrm{X}$-ray tube if the short wavelength limit of the continuous $X$-ray spectrum is equal to $\lambda_{\text {st }}=10.0 \mathrm{pm}$ ?
6.58. A parallel stream of monoenergetic electrons falls normally on a diaphragm with narrow square slit of width $b-1.0 \mu \mathrm{m}$. Find the velocity of the electrons if the width of the central diffraction maximum formed on a screen lecated at a distance $l=50 \mathrm{~cm}$ from the slit is equal to $\Delta x=0.36 \mathrm{~mm}$.
6.59. A parallel stream of electrens accelerated by a potential difference $\boldsymbol{v}=25 \mathrm{~V}$ falls normally on a diaphragm with two narrow slits separated by a distance $d=50 \mu \mathrm{m}$. Calculate the distance between neighbouring maxima of the diffraction pattern on a screen located at a distance $l=100 \mathrm{~cm}$ from the slits.
6.60. A narrow stream of monoenergetie electrons falls at an angle of incidence $\theta=30^{\circ}$ on the natural facet of an aluminium single crystal. The distance between the neighbouring crystal planes parallel to that facet is equal to $d=0.20 \mathrm{~nm}$. The maximum mirror reflection is observed at a certain accelerating veltage $V_{4}$. Find $V_{0}$
252
if the next maximum mirror reflection is known to be observed when the accelerating voltage is increased $\eta=2.25$ times.
6.61. A narrow beam of monoenergetic electrons falls normally on the surface of a $\mathrm{Ni}$ single crystal. The reflection maximum of fourth order is observed in the direction forming an angle $\theta=55^{\circ}$ with the normal to the surface at the energy of the electrons equal to $T-180 \mathrm{eV}$. Calculate the corresponding value of the interplanar distance.
6.62. A narrow stream of electrons with kinetic energy $T=$ – $10 \mathrm{keV}$ passes through a polycrystalline aluminium foil, forming a system of diflraction fringes on a screen. Calculate the interplanar distance corresponding to the rellection of third order from a certain system of crystal planes if it is responsible for a diffraction ring of diameter $D=3.20 \mathrm{~cm}$. The distance between the foil and the screes is $l=10.0 \mathrm{~cm}$.
6.63. A stream of electrons accelerated by a potential difference $y$ falls on the surface of a metal whose inner potential is $V_{1}=15 \mathrm{~V}$. Find:
(a) the refractive index of the metal for the electrons accelerated by a potential difference $V=150 \mathrm{~V}$;
(b) the values of the ratio $V I V_{\text {, at }}$ atich the refractive index diflers from unity by not more than $\eta=1.0 \%$.
6.64. A particle of mass $m$ is located in a unidimensional square potential well with infinitely high walls. The width of the well is equal to 2. Find the permitted values of energy of the particle taking inte account that enly those states of the particle’s motion are realized for which the whole number of de Broglie half-waves are fitted within the given well.
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 $1 \mathrm{mg}$ if the coordiaates of the particles and of the ceatre of the ball are known with uncertainly $1 \mathrm{~mm}$.
6.67. Employing the uncertainty principle, evaluate the indeterminancy of the velocity of an electron in a hydrogen atom if the size of the atom is assumed to be $l=0.10 \mathrm{~nm}$. Compare the obtained magnitude with the velocity of an electron in the first Bohr orbit of the given atom.
6.68. Show that for the particle whose coordinate uncertainty is $\Delta x=\lambda / 2 \pi$, where $\lambda$ is its de Broglie wavelength, the velocity uncertainty is of the same order of magnitude as the particle’s velocity itself.
6.69. A free electron was initially confined within a region with linear dimensions $l=0.10 \mathrm{~nm}$. Using the uncertainty principle, evaluate the time over which the width of the corresponding train of waves becomes $\eta=10$ times as large.
6.70. Employing the uncertainty principle, estimate the minimum kinetic energy of an electron confined within a region whose size is $t=0.20 \mathrm{~nm}$.
6.71. An electron with kinetic energy $T \approx 4 \mathrm{eV}$ is confined within a region whose linear dimension is $l=1 \mu \mathrm{m}$. Using the uncertainty principle, evaluate the relative uncertainty of its veloeity.
6.72. An electron is located in a unidimensional square potential well with infinitely high walls. The width of the well is L. From the uncertainty principle estimate the force with which the electron possessing the minimum permitted energy acts on the walls of the well.
6.73. A particle of mass $m$ moves in a unidimensional potential field $U=k x^{2} / 2$ (harmonic oscillator). Using the uncertainty principle, evaluate the minimum permitted energy of the particle in that field.
6.74. Making use of the uncertainty principle, evaluate the minimum permitted energy of an electron in a hydrogen atom and its corresponding apparent distance from the nucleus.
6.75. A parallel stream of hydrogea atoms with velocity $v=$ $=600 \mathrm{~m} / \mathrm{s}$ falls normally on a diaphragm with a narrow slit behind which a screen is placed at a distance $l=1.0 \mathrm{~m}$. Using the uncertainty principle, evaluate the width of the slit of which the width of its image on the screen is minimum.
6.76. Find a particular solution of the time-dependent Schrōdinger equation for a freely moving partiele of mass $m$.
6.77. A particle in the ground state is located in a unidimensional square potential well of length $l$ with absolutely impenetrable walls $(0<x<D)$. Find the probability of the particle staying within a region $\frac{1}{3} t \leqslant x \leqslant \frac{2}{3} t$.
6.78. A particle is located in a unidimensional square potential well with infinitely high walls. The width of the well is $l$. Find the normalined wave functions of the stationary states of the particle. taking the midpoint of the well for the origin of the $x$ coordinate.
6.79. Demonstrate that the wave functions of the stationary states of a particle confined in a unidimensional potential well with infnitely high walls are orthogonal, i.e. they satisfy the condition $\left\{\hbar_{n} h_{n} d x=0\right.$ if $n^{\prime}
eq n$. Here $l$ is the width of the well, $n$ are integers.
6.80. An electron is located in a unidimensional square potential well with infiaitely high walls. The width of the well equal to $I$ is such that the energy levels are very dense. Find the density of energy levels $d N / d E$, i.e. their number per unit energy interval, as a function of $E$. Calculate $d N / d E$ for $E=1.0 \mathrm{eV}$ if $l=1.0 \mathrm{~cm}$.
6.81. A particle of mass $m$ is located in a two-dimensional square potential well with absolutely impenetrable walls. Find:
$\mathrm{si}$
(a) the particle’s permitted energy values if the sides of the well are $l_{1}$ and $l_{4}$ :
(b) the energy values of the particle at the first four levels if the well has the shape of a square with side $L$.
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 prebability 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 three-dimensional cubie potential well with absolutely impenetrable walls. The side of the cube is equal to 4 . Find:
(a) the preper values of energy of the particle;
(b) the energy diflerence 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.
6.84. Using the Schrüdinger equation, demenstrate 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 lecated in a unidimensional potential field $U(x)$ whose shape is shown in Fig. 6.2, where $U(0)=\infty$. Find:
Fig. 6.2 .
(a) the equation defining the possible values of energy of the particle in the region $E<U_{0}$ : reduce that equation to the form
\[
\sin k l= \pm k l \sqrt{h^{2} / 2 m l^{2} U_{*}}
\]
where $k=\sqrt{2 m E} / h$. 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 ${ }^{2} U_{0}$, at which the first energy level appears in the region $E<U_{0}$. At what minimum value of $H^{2} U_{0}$ does the $n$th level appear?
6.86. Making use of the solution of the foregoing problem, determine the probability of the particle with energy $E=U_{0} / 2$ to be located in the region $x>l$, if $l^{2} U_{0}=\left(\frac{3}{4} \pi\right)^{2} \frac{\hbar^{2}}{m}$.
6.87. Find the possible values of energy of a particle of mass $m$ located in a spherically symmetrical potential well $U(r)=0$ for $r<r_{0}$ and $U(r)=\infty$ for $r=r_{0}$, in the case when the motion of the particle is described by a wave function $\psi(r)$ depending only on $r$.

Instruction. When solving the Schrödinger equation, make the substitution $\psi(r)=\chi(r) / r$.
6.88. From the conditions of the foregoing problem find:
(a) normalized eigenfunctions of the particle in the states for which $\psi(r)$ depends only on $r$;
(b) the most probable value $r_{p r}$ for the ground state of the particle and the probability of the particle to be in the region $r<r_{p r}$.
6.89. A particle of mass $m$ is located in a spherically symmetrical potential well $U(r)=0$ for $r<r_{0}$ and $U(r)=U_{0}$ for $r>r_{0}$.
(a) By means of the substitution $\psi(r)=\chi(r) / r$ find the equation defining the proper values of energy $E$ of the particle for $E<U_{0}$, when its motion is described by a wave function $\psi(r)$ depending only on $r$. Reduce that equation to the form
\[
\sin k r_{0}= \pm k r_{0} \sqrt{\hbar^{2} / 2 m r_{0}^{2} U_{0}}, \text { where } k=\sqrt{2 m E} / \hbar .
\]
(b) Calculate the value of the quantity $r_{0}^{2} U_{0}$ at which the first level appears.
6.90. The wavefunction of a particle of mass $m$ in a unidimensional potential field $U(x)=k x^{2} / 2$ has in the ground state the form $\psi(x)=A \mathrm{e}^{-\alpha x^{2}}$, where $A$ is a normalization factor and $\alpha$ is a positive constant. Making use of the Schrödinger equation, find the constant $\alpha$ and the energy $E$ of the particle in this state.
6.91. Find the energy of an electron of a hydrogen atom in a stationary state for which the wave function takes the form $\psi(r)=$ $=A(1+a r) \mathrm{e}^{-\alpha r}$, where $A, a$, and $\alpha$ are constants.
6.92. The wave function of an electron of a hydrogen atom in the ground state takes the form $\psi(r)=A \mathrm{e}^{-r / r_{1}}$, where $A$ is a certain constant, $r_{1}$ 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.
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 $\psi(r)=A \mathrm{e}^{-r / r_{1}}$, where $A$ is a certain constant, $r_{1}$ is the first Bohr radius.
$35 \cap$
6.94. Particles of mass $m$ and energy $E$ move from the left to the potential barrier shown in Fig. 6.3. Find:
(a) the reflection coefficient $R$ of the barrier for $E>U_{0}$;
(b) the effective penetration depth of the particles into the region $x>0$ for $E<U_{0}$, i.e. the distance from the barrier boundary to the point at which the probability of finding a particle decreases e-fold.
Fig. 6.3.
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:
(a) in Fig. 6.4;
(b) in Fig. 6.5.
Fig. 6.4.
Fig. 6.5.
Fig. 6.6.
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 $U(x)=U_{0}\left(1-x^{2} / l^{2}\right)$