**Exercise 1**

The stationary Schrödinger equation for a particle in a one-dimensional box with infinitely high walls is the equation…

**Answer options:**

one) ;

2 ;

3) ;

4)

**Task 2**

The stationary Schrödinger equation for a particle in a three-dimensional potential box with infinitely high walls is the equation …

**Answer options:**

one) ;

2) ;

3) ;

4)

**Task 3**

The stationary Schrödinger equation in the general case has the form: , where *U* is the potential energy of a microparticle. The linear harmonic oscillator corresponds to the equation…

**Answer options:**

one) ; 2);

3) ; 4)

**Task 4**

The stationary Schrödinger equation for an electron in a hydrogen-like ion is the equation…

**Answer options:**

one) ;

2) ;

3) ;

4)

**Task 5**

The figures show the distribution patterns of the probability density of finding a microparticle in a potential well with infinitely high walls. The state with the quantum number *n* = 2 corresponds to the distribution…

**Answer options:**

one) 2) 3) 4)●

**Task 6**

The figure shows the probability density of detecting a microparticle at various distances from the “walls” of the well. The probability of finding it in the center of the pit is …

**Answer options:**

1) 3/4;

2) 1/2;

thirty;

4) 1/4

**Task 7**

The probability of finding an electron in the section (a,b) of a one-dimensional potential box with infinitely high walls is calculated by the formula , where ω is the probability density determined by the Ψ-function. If the Ψ-function has the form shown in the figure, then the probability of finding an electron in the area equals…

**Answer options:**

1) 1/2;

2) 2/3;

3) 1/3;

4) 5/6

**Task 8**

The probability of finding an electron in the section (a,b) of a one-dimensional potential box with infinitely high walls is calculated by the formula , where ω is the probability density determined by the Ψ-function. If the Ψ-function has the form shown in the figure, then the probability of finding an electron in the area equals…

**Answer options:**

1) 5/8; 2) 3/8; 3) 1/2; 4) 1/4

**Task 9**

The probability of finding an electron in the section *(a,b)* of a one-dimensional potential box with infinitely high walls is calculated by the formula , where ω is the probability density determined by the Ψ -function. If the Ψ function has the form shown in the figure, then the probability of finding an electron in the area equals…

**Answer options:**

1) 3/8; 2) 1/2; 3) 5/8; 4) 1/4

**Task 10**

The probability of finding an electron in the section *(a,b)* of a one-dimensional potential box with infinitely high walls is calculated by the formula , , where *ω* is the probability density determined by the *ψ* function. If *ψ –* the function has the form shown in the figure, then the probability of finding an electron in the area equals…

**Answer options:**

1) 2/3;

2) 5/6; 3265

3) 1/2;

4) 1/3.

**Task 11**

The electron is in an excited state ( *n* =2) in a one-dimensional potential box of width *a* with infinitely high walls. The probability density of finding an electron is maximum at points with coordinates …

**Answer options:**

one) ;

2) ;

3) ;

4)

**Task 12**

The electron is located in the first third of a rectangular one-dimensional potential box with impenetrable walls at the second energy level. The probability of finding an electron in the center of this potential box at the same energy level is …

**Answer options:**

1) 0.1;

2) 0.3;

3) 0.7;

4)0;

5)0.5

**Task 13**

The wave function of a particle in a potential well with infinitely high walls of width L has the form: . The momentum of a particle in the first excited state ( *n* =2) is …

**Answer options:**

one) 3)

2) 4)

**Task 14**

The wave function of a particle in a potential well with infinitely high walls of width L has the form: If the momentum of the particle is then the particle is at the energy level with the number …

**Answer options:**

1) *n* =4; 2) *n* =2; 3) *n* =1; 4) *n* =3

**Task 15**

Wave function of a particle in a potential well with infinitely high walls and width L has the form:. If the momentum of the particle is , then the de Broglie wavelength of this particle is …

**Answer options**

one) 2) 3) 4) *3L*

**Task 16**

An electron is located in a potential well of infinite depth. wave functions

are schematically shown in the figure. Which of these states will be preserved if the width of the potential well is halved?

*a) 1,3,5;*

*b) 2,4,6;*

*c) only 1;*

*d) 1,2,3,4,5,6;*

*e) there is no correct answer.*

**Task 17**

An electron with kinetic energy and moving from left to right, meets on the way in one case a threshold (P), and in the other – a barrier (B) with a height in both cases. From the point of view of classical and quantum theory, the probability of overcoming the threshold by an electron and barrier is different and depends on the ratio and . Match and fill in the table:

*a) ; b) ; in) ; G) ; e)*

**Task 18**

Choose the correct answer for the units of the one-dimensional psi-function (ψ=ψ(х))

a) dimensionless;

b) ;

in) ;

G) ;

d) there is no correct answer

**Task 19**

An electron is located in a potential well with vertical walls. Its wave function is shown in the figure. In this case, the depth of the potential well

a) is endless;

b) infinite on the left, finite on the right;

c) infinite on the right, finite on the left;

d) finite;

e) there is no correct answer.

**Task 20**

The physical meaning of the psi-function is that

a) its modulus describes the motion of a particle;

b) it shows the probability density of finding a particle in the vicinity of a given point in space;

c) the square of its modulus shows the probability density of finding a particle in the vicinity of a given point in space;

d) the cube of its modulus shows the probability of finding a particle at a given point in space;

e) there is no correct answer.

**Task 21**

The electron is in a potential well with infinitely high walls. For some states in the middle of the well, the psi-function of an electron can have a node, i.e. y=0. Choose the correct statement:

a) the psi-function cannot have nodes in a well with infinite walls;

b) the psi-function cannot have a node in the center of the pit;

c) state numbers are multiples of two;

d) state numbers are multiples of three.

**Task 22**

The particle is in a rectangular one-dimensional potential well with infinitely high walls in a state with the main quantum number n.

a) n;

b) n+1;

c) n+2;

d) n+3;

e) there is no correct answer.

**Task 23**

Choose the correct continuation of the sentence. For macroscopic bodies, for example, a speck of dust in a matchbox, we do not notice the quantization of energy levels, because

a) macroscopic bodies do not obey the laws of quantum mechanics;

b) the energy levels of macroscopic bodies are located so rarely that the quantization of energy is not noticeable;

c) the energy levels of macroscopic bodies are located so often that the energy quantization is not noticeable;

d) experiments to detect the quantization of the energy of macroscopic bodies have not been carried out.

**Task 24**

An electron is located in a potential well of infinite depth. The wave functions are schematically presented in the figure. The hole was halved. How many times will the minimum value of the kinetic energy of the electron change in this case?

a) will not change

b) 2 times;

c) 4 times;

d) 6 times;

e) no

correct answer.

**Task 25**

The particle is located in a one-dimensional infinitely deep rectangular potential well. Match the Dependency Plot and status number . Fill the table:

State number | corresponding letter |

**Task 26**

The normalization condition for the psi-function for a particle located in a potential well with impenetrable walls of width l is that the probability of finding a particle inside the well is equal to:

a) 0;

b) 1/l;

in 1;

G) ;

e) there is no correct answer.

**Task 27**

Choose the wrong statements

a) the Schrödinger equation describes the motion of a quantum particle;

b) the Schrödinger equation can be obtained by refining Newton’s laws in classical mechanics;

c) quantum theory insists on the rejection of absolute certainty in setting the initial conditions for the motion of a particle;

d) in quantum theory, only the real part of the complex wave function has physical meaning;

e) for macroscopic particles, the predictions of quantum and classical theory coincide.

**Task 28**

The particle is located in a rectangular one-dimensional potential well with impenetrable walls. The general solution for the stationary Schrödinger equation has the form:

a) ;

b) ;

in) ;

G) ;

e) no correct solutions are given.

**Task 29**

The figures schematically show the dependences of the particle detection probability density on the coordinate. Establish a correspondence between the shape of a one-dimensional rectangular potential well and the figure and fill in the table.

Potential well shape | corresponding letter |

Pit walls of finite height | |

Both walls of finite height, right wall higher | |

Both walls of final height, the left wall is higher | |

Left wall of finite height, right wall of infinite height | |

Right wall of finite height, left wall of infinite height | |

Walls of a pit of infinite height |

**Task 30**

A particle of mass m is located in a one-dimensional rectangular well of width l with impenetrable walls in a state with psi-function Y _{n} (x,t) and energy spectrum . What is the number of wave function nodes inside the well in the region 0

a) n-1;

b) n;

c) n+1;

d) n+2;

e) there is no correct answer.

**Task 31**

The particle is located in a one-dimensional rectangular well of width l with impenetrable walls in various states. In which of the following states does its energy have a certain value?

a) y(x)=Ax(lx);

b) y(x)=Asin(px/l);

c) y(x)= Asin (px/l) + Вsin (3px/l)

d) there is no correct answer.

**THEME 3**

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