Introduction to Atomic and Quantum Physics - Schroedinger Equation Independent of Time

7/30/2019 Introduction to Atomic and Quantum Physics - Schroedinger Equation Independent of Time

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Lectures:

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 11

Chapter 12

X.

Schrdinger equation LIEU OF ALL TIME

Independent Schrdinger equation since this equation as:

The regularity conditions and :

a. Finiteb. continuousc. clear

Form (shape) of the function depends on its potential V. a) For the free particle V (x) = 0 equation (X.1) reduces to the form:

Size k in formula (X.4) is equal to:

{E} - a set of continuous

b) particle in an infinitely deep potential well

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(X.1)

(X.2)

(X.3)

(X.4)

(X.5)
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Rys.X.1. Infinitely deep potential well. The particle has no right to stay in areas I and III due to the huge potential barrier.

Area II:

V (x) = 0,

Equation (X.6) as for the harmonic oscillator.

(X.7a) and (X.7b) are two specific solutions of the equation (6).

Eigenfunction y 2 (x) does not satisfy the condition of continuity because: - the lack of continuity

By contrast, the self-y 1 (x) is satisfied for this condition:

From formulas (X.5) and (X.8) we get that energy to the n-th energy level is given by:

From formula (X.9) shows that the set of energy {E} is discrete, so quantization.features its own particles confined in the cavity potential:

(X.6)

(X.7a)

(X.7b)

(X.8)

(X.9)

(X.10)


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Rys.X.2. Graphic illustration of (X.9).

Rys.X.3. Graphic illustration of (X.10).


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ys. . . ar pro a y ens y or eren n .

X.1. FORM operators Schrdinger equation.

{A}:

a set of continuous (free particle),discrete (particle in the cavity potential)

If the value of their own to be more than one eigenfunction that the value is degenerate.

If a and n are different eigenfunctions {y 1 , y2 , ..., y n } is the n - fold degeneracy (degeneracy)

Record shall be equivalent to equation (X.1.1):

where: - Hamiltonian (Hamiltonian operator) is expressed by the formula:

Expression (X.1.3) will also be recorded in the abridged version:

where:

the Laplace operator

X.2. ENERGY OPERATOR.

Energy operator is called the expression:

Equation own energy operator is of the form:

that is:

X.3. OPERATOR momentum.

(X.1.1)

(X.1.2)

(X.1.3)

(X.1.4)

(X.1.5)

(X.2.1)

(X.2.2)

(X.2.3)


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We are looking for the operator:

After dividing equation (X.3.2) by we get:

The property operators:

From equations (X.3.3) and (X.3.4) shows that:

Similarly, operators can be found:

X.4. FEATURES OF YOUR OWN AND SOLE angular momentum

The old quantum theory:

Second postulate ofBohr :

Wilson quantization rules - Somerfelda:

Constituent angular momentum L assign their respective operators:

(X.4.4a)

(X.3.1)

(X.3.2)

(X.3.3)

(X.3.4)

(X.3.5a)

(X.3.5b)

(X.3.5c)

(X.4.1)

(X.4.2)

(X.4.3)

(X.4.5a)
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(X.4.4b)

(X.4.4c)

Substituting the formulas (X.4.5) obtained earlier component of the momentum operators we get:

Equation (X.4.7) after substituting the value of the component of angular momentum operator (X.4.6c) takes the form:

Polar coordinates:

Angular momentum operator in polar coordinates:

The equation of its own - cond component:

(X.4.5b)

(X.4.5c)

(X.4.6a)

(X.4.6b)

(X.4.6c)

(X.4.7)

(X.4.7a)

(X.4.8a)

(X.4.8b)


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From formulas (X.4.7) and (X.4.8c) follows:

Equation (X.4.13) is a mathematical solution to the equation own (X.4.7).

Assumptions:

A = 1

The uniqueness of the function, with this assumption, we find eigenfunctions.

where m - magnetic quantum number,

angular momentum components are subject to uncertainty principle of Heisenberg.

X.5. FEATURES OF YOUR OWN AND SOLE

(X.4.8c)

(X.4.9)

(X.4.10)

(X.4.11)

(X.4.12)

(X.4.13)

(X.4.14)

(X.4.15)

(X.5.1)

(X.5.2)
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- Eigenfunction

We use the method of separation of variables:

From formulas (X.5.4) and (X.5.5) we get:

In the formula (X.5.6) will be equal to the left side of the right if and only if both sides of the equation will be fixed:

Solution (X.5.7b) exists if and only if

l = 0,1,2, ...

Quantization of L is different than the old quantum theory. According to her, the angular momentum is given by:

,

L L * for large l The biggest difference in the values of angular momentum is the minimum value.From equation (X.5.10) shows that the minimum value of the angular momentum is equal to:

However, according to the old quantum theory, the minimum value of angular momentum:

So we have a contradiction, because:

(X.5.3)

(X.5.4)

(X.5.5)

(X.5.6)

(X.5.7a)

(X.5.7b)

(X.5.8)

(X.5.9)

(X.5.10)


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The experiment confirms that the relationship (X.5.11) is true:

where - Legendre polynomial

l | M |

0 0 1

11

10

1

222

210

3

Table X.1. Legendre polynomial sample values for different values of the quantum numbers l and m

From expressions (X.5.5), (X.5.6) and (X.5.11) we get that:

Orbital quantum number l determines the electronic states.

l 0 1 2 3 ...

Status symbol s p d f ...

Table X.2. El ectron states for the corresponding values of l

S electron, the one for which the orbital angular momentum is equal to 0, the electron p - orbital angular momentum equal to 1,etc.

X.6. Free particle wave function (MATTER WAVES).

Free particle - the potential V is equal to 0

V (x, y, z) = 0

assumption 1:

After substituting expressions (X.6.2) to the equation (X.6.1) we get:

Features its own data model:

Assumption 2:

(X.5.11)

(X.5.12)

(X.6.1)

(X.6.2)

(X.6.3)

(X.6.4)


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A = 1

Calculated that the ratio is:

The three-dimensional model (X.6.6) takes the form:

The character of the wave function:

a. in one dimension (1D):

b. In three dimensions (3D):

According to the result of the probability of finding the particle is the same everywhere, which contradicts the definition ofparticles, because the particle is in a certain place, and not everywhere. A better solution for a free particle wave packet,including solves the problem of location.

X.7. PACKAGE wave.

The definition of a wave:the wave function is that at some point (area) has a value different from zero, and after that the area is equal to 0

The construction of the wave packet:

1D

c (k 0 ) - amplitude function.

Wave function (X.7.1) when expanded in series is:

(X.6.5)

(X.6.6)

(X.6.7)

(X.6.8)

(X.6.9)

(X.7.1)


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Expression (X.7.2) is the mathematical form of the wave packet. They can be written as:

Where c (x, t) is the amplitude, and is given by:

where:

Because they must be met:

and

From the first we obtain that:

However, the second:

Rys.X.2. The dependence of the wave function of the position x

(X.7.2)

(X.7.3)

(X.7.4)

(X.7.5)


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Rys.X.3. Probability of finding the particle in the interval [- X, + X] of x

X.8. Group velocity u

Group velocity is the speed at which the maximum of the main moves in the wave packet. It is equal to the wave particle velocityde Broglie.

Wave phase velocity - a velocity at which thephase shifts, eg, point 1

Equation (X.8.5) shows the relationship between the phase velocity v, and the group velocity uThese values are the same when the speed is independent of wavelength (no dispersion).

X.9. SPEED REPORT GROUP (u) the particle velocity (v 0 ).

Description classical particles:

(X.8.1)

(X.8.2)

(X.8.3)

(X.8.4)

(X.8.5)
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Description of the same particles by waves of matter:
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Introduction to Atomic and Quantum Physics - Schroedinger Equation Independent of Time