Schrödinger equation for a particle in a potential field. Schrödinger equation

Movement of microparticles in various force fields is described within the framework of non-relativistic quantum mechanics using the Schrödinger equation, from which experimentally observable conditions follow wave properties particles. This equation, like all the basic equations of physics, is not derived, but postulated. Its correctness is confirmed by the agreement of the calculation results with experience. The Schrödinger wave equation has the following general view :

- (ħ 2 / 2m) ∙ ∆ψ + U (x, y, z, t) ∙ ψ = i ∙ ħ ∙ (∂ψ / ∂t)

where ħ = h / 2π, h = 6.623∙10 -34 J ∙ s - Planck’s constant;
m is the particle mass;
∆ - Laplace operator (∆ = ∂ 2 / ∂x 2 + ∂ 2 / ∂y 2 + ∂ 2 / ∂z 2);
ψ = ψ (x, y, z, t) - the desired wave function;
U (x, y, z, t) - potential function particles in the force field where it moves;
i is the imaginary unit.

This equation has a solution only under the conditions imposed on the wave function:

1. ψ (x, y, z, t) must be finite, single-valued and continuous;
2. its first derivatives must be continuous;
3. function | ψ | 2 must be integrable, which in the simplest cases reduces to the condition for normalizing the probabilities.

∆ψ + (2m / ħ 2) ∙ (E - U) ∙ ψ = 0

where ψ = ψ (x, y, z) is the wave function of coordinates only;
E - equation parameter - total energy particles.

For this equation, only those solutions that are expressed by regular functions ψ (called own functions), which occur only for certain values ​​of the parameter E, called the energy eigenvalue. These E values ​​can form either a continuous or discrete series, i.e. both continuous and discrete energy spectrum.

For any microparticle, in the presence of a Schrödinger equation of type (8.2), the problem of quantum mechanics is reduced to solving this equation, i.e. finding the values ​​of the wave functions ψ = ψ (x, y, z), corresponding to the spectrum of intrinsic energies E. Next, find the probability density | ψ | 2, which in quantum mechanics determines the probability of finding a particle in a unit volume in the vicinity of a point with coordinates (x, y, z).

One of the simplest cases of solving the Schrödinger equation is the problem of the behavior of a particle in a one-dimensional rectangular “potential well” with infinitely high “walls”. Such a “hole” for a particle moving only along the X axis is described by potential energy of the form

where l is the width of the “hole”, and the energy is measured from its bottom (Fig. 8.1).

The Schrödinger equation for stationary states in the case of a one-dimensional problem will be written in the form:

∂ 2 ψ / ∂x 2 + (2m / ħ 2) ∙ (E - U) ∙ ψ = 0

Due to the fact that the “walls of the pit” are infinitely high, the particle does not penetrate beyond the “pit”. This leads to the boundary conditions:

ψ (0) = ψ (l) = 0

Within the “well” (0 ≤ x ≤ l), equation (8.4) reduces to the form:

∂ 2 ψ / ∂x 2 + (2m / ħ 2) ∙ E ∙ ψ = 0

∂ 2 ψ / ∂x 2 + (k 2 ∙ ψ) = 0

where k 2 = (2m ∙ E) / ħ 2

ψ (x) = A ∙ sin (kx)

where k = (n ∙ π)/ l

for integer values ​​of n.

From expressions (8.8) and (8.10) it follows that

E n = (n 2 ∙ π 2 ∙ ħ 2) / (2m ∙ l 2) (n = 1, 2, 3 ...)

Consequently, a microparticle in a “potential well” with infinitely high “walls” can only be at a certain energy level E n , i.e. in discrete quantum states n.

Substituting expression (8.10) into (8.9) we find the eigenfunctions

ψ n (x) = A ∙ sin (nπ / l) ∙ x

which for this case will be written as:

From where, as a result of integration, we obtain A = √ (2 / l) and then we have

ψ n (x) = (√ (2 / l)) ∙ sin (nπ / l) ∙ x (n = 1, 2, 3 ...)

The graphs of the function ψ n (x) have no physical meaning, while the graphs of the function | ψ n | 2 show the distribution of the probability density of detecting a particle at various distances from the “walls of the pit” (Fig. 8.1). It is these graphs (as well as ψ n (x) - for comparison) that are studied in this work and clearly show that the ideas about particle trajectories in quantum mechanics are untenable.

From expression (8.11) it follows that the energy interval between two neighboring levels is equal to

∆E n = E n-1 - E n = (π 2 ∙ ħ 2) / (2m ∙ l 2) ∙ (2n + 1)

From this it is clear that for microparticles (such as electrons) at large sizes"holes" (l≈ 10 -1 m), the energy levels are located so closely that they form an almost continuous spectrum. This condition occurs, for example, for free electrons in metal. If the dimensions of the “well” are comparable to atomic ones (l ≈ 10 -10 m), then a discrete energy spectrum is obtained ( line spectrum). These types of spectra can also be studied in this work for various microparticles.

Another case of the behavior of microparticles (as well as microsystems - pendulums), often encountered in practice (and considered in this work), is the problem of a linear harmonic oscillator in quantum mechanics.

As is known, potential energy one-dimensional harmonic oscillator of mass m is equal to

U (x) = (m ∙ ω 0 2 ∙ x 2)/ 2

where ω 0 is the natural frequency of oscillator oscillator ω 0 = √ (k / m);
k is the elasticity coefficient of the oscillator.

Dependence (8.17) has the form of a parabola, i.e. "potential hole" in in this case is parabolic (Fig. 8.2).

ψ n (x) = (N n ∙ e -αx2 / 2) ∙ H n (x)

where N n is a constant normalizing factor depending on the integer n;
α = (m ∙ ω 0) / ħ;
H n (x) is a polynomial of degree n, the coefficients of which are calculated using a recurrent formula for different integer n.
In theory differential equations it can be proven that the Schrödinger equation has a solution (8.18) only for the energy eigenvalues:

E n = (n + (1 / 2)) ∙ ħ ∙ ω 0

This means that the energy of a quantum oscillator can only take discrete values, i.e. quantized. When n = 0, E 0 = (ħ ∙ ω 0) / 2 takes place, i.e. zero-point energy, which is typical for quantum systems and represents direct consequence uncertainty relations.

As a detailed solution of the Schrödinger equation for a quantum oscillator shows, each eigenvalue of energy for different n corresponds to its own wave function, because the constant normalizing factor depends on n

and also H n (x) - Chebyshev-Hermite polynomial of degree n.
Moreover, the first two polynomials are equal:

H 0 (x) = 1;
H 1 (x) = 2x ∙ √ α

Any subsequent polynomial is related to nmi according to the following recurrent formula:

H n+1 (x) = 2x ∙ √ α ∙ H n (x) - 2n ∙ H n-1 (x)

Eigenfunctions of type (8.18) allow us to find for a quantum oscillator the probability density of finding a microparticle as | ψ n (x) | 2 and examine its behavior on various levels energy. Solving this problem is difficult due to the need to use a recurrent formula. This problem can be successfully solved only using a computer, which is what is being done in this work.

I am giving this lecture to you for entertainment. I wanted to see what would happen if I started reading in a slightly different style. It is not included in the course, and do not think that this is an attempt to teach you how to last hour something new. I rather imagine that I am giving a seminar or presenting a research report in front of a more advanced audience, in front of people who already understand a lot about quantum mechanics. The main difference between a seminar and a regular lecture is that at a seminar the speaker does not present all the stages, the entire algebra of calculations. He simply says: “If you do such and such, this is what you will get,” but does not go into detail. So in this lecture we will only express ideas and present the results of calculations. And you must understand that it is not at all necessary to immediately and completely understand everything, you just need to believe that if you do all the calculations, then everything will work out.

But that's not all. The main thing is that I want to talk about it. This is so fresh, relevant, modern theme, that it is quite legal to bring it to the seminar. This topic is a classic aspect of the Schrödinger equation, the phenomenon of superconductivity.

Typically, the wave function that appears in the Schrödinger equation applies to only one or two particles. And the wave function itself does not have a classical meaning, unlike the electric field, or vector potential, or other similar things. True, the wave function of an individual particle is a “field” in the sense that it is a function of position, but, generally speaking, it does not have a classical meaning. However, there are sometimes circumstances in which the quantum mechanical wave function actually has classical meaning, that’s exactly what I want to touch on. The peculiarity of the quantum mechanical behavior of matter on small scales usually does not make itself felt in large-scale phenomena, except for the standard conclusions that it gives rise to Newton's laws, the laws of the so-called classical mechanics. But there are sometimes circumstances in which the features of quantum mechanics can have a special effect on large-scale phenomena.

At low temperatures, when the energy of the system decreases very, very strongly, instead of the previous huge number of states, only a very, very small number of states are included in the game - those that are located not far from the main one. Under such conditions, the quantum mechanical character of this ground state can manifest itself at the macroscopic level. It is the purpose of this lecture to demonstrate the connection between quantum mechanics and large-scale effects - not the usual discussion of the way in which quantum mechanics reproduced on average Newtonian mechanics, but a special case when quantum mechanics causes its own, characteristic effects on large, “macroscopic” dimensions.

Let me start by reminding you of some properties of the Schrödinger equation. I want to use the Schrödinger equation to describe the behavior of a particle in a magnetic field, because the phenomena of superconductivity are associated with magnetic fields. An external magnetic field is described by a vector potential, and the question is what are the laws of quantum mechanics in the field of a vector potential. The principle that determines the quantum mechanical behavior of a particle in a vector potential field is very simple. The amplitude that a particle, in the presence of a field, will move along a certain path from one place to another (Fig. 19.1) is equal to the amplitude that it would pass along this path without the field, multiplied by the exponent of curvilinear integral from the vector potential, multiplied in turn by electric charge and divided by Planck's constant [see Ch. 15, § 2 (issue 6)]:

This is the original statement of quantum mechanics.

Fig. 19.1. The amplitude of the transition from to along the path is proportional .

And in the absence of a vector potential, the Schrödinger equation for a charged particle (non-relativistic, without spin) has the form

where is the electric potential, so is the potential energy. And equation (19.1) is equivalent to the statement that in a magnetic field the gradients in the Hamiltonian must be replaced each time by the gradient minus , so that (19.2) turns into

This is the Schrödinger equation for a particle with a charge (non-relativistic, without spin) moving in an electromagnetic field.

To make it clear that it is correct, I want to illustrate this with a simple example, where instead of a continuous case there is a line of atoms placed on an axis at a distance from each other, and there is an amplitude for the electron to jump in the absence of a field from one atom to another. Then, according to equation (19.1), if there is a vector potential in the -direction, then the jump amplitude will change compared to what was before, it will have to be multiplied by - an exponent with an indicator, equal to the product on vector potential, integrated from one atom to another. For simplicity we will write , since, generally speaking, depends on . If we denote by the amplitude that an electron is found near an atom located at a point, then the rate of change of this amplitude will be given by the equation

It has three parts. Firstly, the electron that is at the point has some energy. This, as usual, gives a member. Then there is a term, i.e., the amplitude of the fact that an electron from an atom located at jumped back a step. However, if this occurs in the presence of a vector potential, then the phase of the amplitude must shift according to rule (19.1). If the distance between neighboring atoms does not change noticeably, then the integral can be written simply as the value in the middle multiplied by the distance. So the product times the integral is equal to . And since the electron jumped back, I mark this phase shift with a minus sign. This gives the second part. And in the same way, there is a certain amplitude that there will be a jump forward, but this time the vector potential is taken on the other side of , at a distance, and multiplied by the distance. This gives the third part. In sum, we obtain an equation for the amplitude that a particle in a field characterized by a vector potential will end up at point .

But further we know that if the function is smooth enough (we take the long-wave limit) and if we move the atoms closer together, then equation (14.4) (p. 80) will approximately describe the behavior of the electron in vacuum. Therefore, the next step is to expand both sides of (19.4) in powers, considering it to be very small. For example, if , then right side will be equal to simply , so in the zeroth approximation the energy is equal to . Then the powers will come, but due to the fact that the signs of the exponents are opposite, only even powers will remain. As a result, if you expand the Taylor series , and exponentials and then collect the terms with , you get. Now remember that solutions in zero magnetic field (see Chapter I, § 3) depict a particle with effective mass given by the formula

If you then put it down and come back to , then you can easily verify that (19.6) is the same as the first part of (19.3). (The origin of the potential energy term is well known, and I will not go into it.) The statement (19.1) that the vector potential multiplies all amplitudes by an exponential factor is equivalent to the rule that the momentum operator is replaced by , as we did in the Schrödinger equation (19.3).

SCHRÖDINGER EQUATION
AND ITS SPECIAL CASES (continued): passage of a particle through a POTENTIAL BARRIER, Harmonic oscillator

Passage of a particle through a potential barrier for the classical case we have already considered in LECTURE 7 OF PART 1 (see Fig. 7.2). Let us now consider a microparticle whose total energy is less than the level U potential barrier (Fig. 19.1). IN classic version in this case, the passage of the particle through the barrier is impossible. However, in quantum physics there is a possibility that the particle will pass through. Moreover, it will not “jump over” it, but will, as it were, “leak through”, using its wave qualities. Therefore, the effect is also called “tunnel”. For each of the areas I, II, III let's write down stationary equation Schrödinger (18.3).

For I And III: , (19.1, a)

For II: https://pandia.ru/text/78/010/images/image005_107.gif" width="71" height="32">, where a = const. Then and y" = . Substitution of y" into (19.1a) gives: The required general solution for the region I will be written as a superposition

https://pandia.ru/text/78/010/images/image010_62.gif" width="132" height="32 src="> . (19.3)

In this case starting point wave propagation is shifted by L,a IN 3 = 0 , since in the area III there is only a passing wave.

In the area II(barrier) substitution of y" in (19.1b) gives

https://pandia.ru/text/78/010/images/image012_51.gif" width="177" height="32">.

The probability of passage is characterized pass rate- the ratio of the intensity of the transmitted wave to the intensity of the incident one:

(0) = y2"(0) , y2"( L) = y3"( L); (19.5)

of which the first two mean the “stitching” of functions on the left and right boundaries of the barrier, and the third and fourth mean the smoothness of such a transition. Substituting the functions y1, y2 and y3 into (19.5), we obtain the equations

Let's divide them into A 1 and denote a 2=A 2/A 1; b 1=B 1/A 1; a 3=A 3/A 1; b 2=B 2/A 1.

. (19.6)

Let's multiply the first equation (19.6) by ik and add it with the second one. We get 2 ik = a 2(q +ik)-b 2(q-ik) . (19.7)

We will consider the second pair of equations (19.6) as a system of two equations with unknowns a 2 and b 2.

Determinants of this system:

https://pandia.ru/text/78/010/images/image017_33.gif" width="319" height="32">,

where e- qL(q+ik) 2 » 0, because qL >> 1.

Therefore https://pandia.ru/text/78/010/images/image019_32.gif" width="189" height="63">, and to find the modulus of a complex value A 3, multiply the numerator and denominator of the resulting fraction by ( q +ik)2. After simple transformations we get

https://pandia.ru/text/78/010/images/image021_30.gif" width="627" height="135 src=">Usually E/U~ 90% and the entire coefficient before “e” is of the order of one. Therefore, the probability of a particle passing through the barrier is determined by the following relation:

https://pandia.ru/text/78/010/images/image023_24.gif" width="91" height="44">.

This means that when E< U the particle will not overcome the barrier, i.e. the tunnel effect in classical physics absent.

This effect is used in engineering practice to create tunnel diodes, widely used in radio devices (see PART 3, LECTURE 3).

In addition, it turned out to be possible to initiate terrestrial conditions thermonuclear reaction synthesis, which occurs on the Sun under normal conditions for the Sun - at a temperature T ~ 109 K. There is no such temperature on Earth, however, thanks to tunnel effect, there is a probability that the reaction will start at temperature T ~ 107 K that occurs during an explosion atomic bomb, which was the ignition device for the hydrogen one. More about this in the next part of the course.

Harmonic oscillator.Classical We have also already considered the harmonic oscillator (LECTURES 1,2 PART 3). For example, they are spring pendulum, whose total energy E = mV 2/2 + kx 2/2. Theoretically, this energy can take on a continuous series of values, starting from zero.

A quantum harmonic oscillator is an oscillating harmonic law microparticle located in bound state inside an atom or nucleus. In this case, the potential energy remains classical, characterizing a similar elastic restoring force kx. Considering that the cyclic frequency we obtain for potential energy https://pandia.ru/text/78/010/images/image026_19.gif" width="235" height="59">. (19.9)

Mathematically, this problem is even more complex than the previous ones. Therefore, we will limit ourselves to stating what will happen as a result. As in the case of a one-dimensional well, we get discrete spectrum of eigenfunctions and eigenenergies, and one eigenvalue of energy will correspond to one wave function: EnÛ y n(there is no degeneracy of states, as in the case of a three-dimensional well). The probability density |yn|2 is also an oscillating function, but the height of the “humps” is different. It's no longer trivial sin2 , and the more exotic Hermite polynomials Hn(x). Wave function looks like

, Where WITHn- depending on n constant. Energy eigenvalue spectrum:

, (19.10)

where is the quantum number n = 0, 1, 2, 3 ... . Thus, there is also " zero energy" , above which the energy spectrum forms a “shelf”, where the shelves are located at the same distance from each other (Fig. 19.2). The same figure shows for each energy level the corresponding probability density |yn|2, as well as the potential energy external field(dotted parabola).

The existence of a nonzero minimum possible oscillator energy has deep meaning. This means that the vibrations of microparticles do not stop never, which in turn means unattainability absolute zero temperature.

1. , Bursian physics: A course of lectures with computer support: Proc. aid for students higher textbook institutions: In 2 volumes - M.: Publishing house VLADOS-PRESS, 2001.

In principle, nothing special, they can be found in tables and even graphs.

Let the particle move along the X axis. In this case, the movement is limited by the segment ( 0.l). At points x=0 and x=l there are impenetrable infinitely high walls. The potential energy in this case has the form

This dependence of potential energy on x is called potential well.

Let us write the stationary Schrödinger equation

Since the psi function depends only on the x coordinate, the equation simplifies as follows

Inside the potential well U=0

The particle cannot go beyond the potential well. Therefore, the probability of detecting a particle outside the well is zero. Accordingly, the psi-function outside the hole is equal to zero. From the continuity condition it follows that ψ must be equal to zero at the boundaries of the well, i.e. . This is the boundary condition that solutions to the equation must satisfy.

Let us introduce the notation

and we obtain an equation well known from the theory of oscillations

The solution to such an equation has the form harmonic function

The choice of the corresponding parameters k and α is determined by the boundary conditions, namely,

n = 0 is eliminated, because in this case ψ = 0 and the particle is nowhere. Consequently, the number k takes only certain discrete values ​​that satisfy the condition. It follows very important result. We'll find eigenvalues particle energy

those. The energy of an electron in a potential well is not arbitrary, but takes discrete values, i.e. is quantized. The value of E n depends on the integern , which takes a value from 1 to ∞ and is called main quantum number . Quantized energy values ​​are called energy levels and the quantum number n determines the energy level number. Thus, an electron in a potential well can be at a certain energy level E n. Moreover, the minimum energy value corresponding to the first energy level is different from zero

.

Let's determine the distance between adjacent energy levels

At large m and l, the distance between the levels becomes small and the spectrum becomes quasi-continuous. Relative distance between levels

as n → ∞,

i.e., the spectrum becomes continuous. This is Bohr's correspondence principle: For large quantum numbers, the conclusions and results of quantum mechanics must agree with the classical results.

Let's return to the problem of determining eigenfunctions. After applying the boundary conditions we have

To find the coefficient A, we use the normalization condition

The value of the integral is l /2.

Thus, the eigenfunctions have the form

The graphs of eigenfunctions look like

Let us finally formulate main conclusions:

1. The energy spectrum of a particle in a potential well is discrete - the energy is quantized.

2. Minimum value kinetic energy cannot be equal to zero.

3. Discrete nature energy levels appears at low m,l And n, at large m,l,n the movement becomes classic.

4. The positions of a microparticle in the well are not equally probable, but are determined by their own functions, while in the case of a classical particle all positions are equally probable.

Questions for self-control:

1. How to determine the probability of finding a particle at a certain point?

2. What is called a potential well?

3. What is the meaning of the Schrödinger equation? What does Schrödinger's equation allow us to find?

4. What conditions are imposed on the psi function?

5. What is the physical meaning of the principal quantum number?

6. Why is quantum mechanics a statistical theory?

7. What is Bohr's correspondence principle?

For particles quantum world other laws apply than for objects of classical mechanics. According to de Broglie's assumption, microobjects have the properties of both particles and waves - and, indeed, when an electron beam is scattered by a hole, diffraction characteristic of waves is observed.

Therefore, we can not talk about movement quantum particles, but about the probability that the particle will be in specific point at some point in time.

What does the Schrödinger equation describe?

The Schrödinger equation is intended to describe the features of the movement of quantum objects in fields external forces. Often a particle moves through a force field that does not depend on time. For this case, the stationary Schrödinger equation is written:

In the presented equation, m and E are and, accordingly, the energy of a particle located in a force field, and U is this field. — Laplace operator. — Planck’s constant equal to 6.626 10 -34 J s.

(it is also called the probability amplitude, or psi-function) - this is a function that allows us to find out in which place in space our microobject will most likely be located. It is not the function itself that has the physical meaning, but its square. Probability that a particle is in an elementary volume:

Therefore, it is possible to find a function in a finite volume with the probability:

Since the psi function is a probability, it cannot be either less than zero, nor exceed one. Total probability finding a particle in an infinite volume is a normalization condition:

The principle of superposition works for the psi function: if a particle or system can be in a number of quantum states, then a state determined by their sum is also possible for it:

The stationary Schrödinger equation has many solutions, but when solving it should be taken into account boundary conditions and select only own solutions- those who have physical meaning. Such solutions exist only for individual values energy of the particle E, which form the discrete energy spectrum of the particle.

Examples of problem solving

EXAMPLE 1

EXAMPLE 2