phys 342 modern physics 2. course elements 1.the schrödinger equationthe schrödinger equation...

PHYS 342

Modern Physics 2

Course Elements

1. The Schrödinger Equation2. The Rutherford-Bohr Model of the Atom3. The Hydrogen Atom in Wave Mechanics4. Many-Electron Atoms5. Molecular Structure6. Statistical Physics7. Nuclear Structure and Radioactivity

1. The Schrödinger Equation

a. Justification of the Schrödinger Equationb. The Schrödinger Recipec. Probabilities and Normalizationd. Applicationse. The Simple Harmonic Oscillatorf. Time Dependenceg. Steps and Barriers

2. The Rutherford-Bohr Model of the Atom

a. Basic Properties of Atomsb. The Thomson Modelc. The Rutherford Nuclear Atomd. Line Spectrae. The Bohr Modelf. The Franck-Hertz Experiment

3. The Hydrogen Atom in Wave Mechanics

a. The Schrödinger Equation in Spherical Coordinates

b. The Hydrogen Atom Wave Functionsc. Radial Probability Densitiesd. Angular Momentum and Probability Densitiese. Intrinsic Spinf. Energy Levels and Spectroscopic Notationg. The Zeeman Effect

4. Many-Electron Atoms

a. The Pauli Exclusion Principleb. Electronic States in Many-Electron Atomsc. The Periodic Tabled. Properties of the Elementse. X-Raysf. Optical Spectrag. Lasers

5. Molecular Structure

a. The Hydrogen Molecule Ionb. The H2 Molecule an the Covalent Bond

c. Other Covalent Bonding Moleculesd. Ionic Bondinge. Molecular Vibrationsf. Molecular Rotationsg. Molecular Spectra

6. Statistical Physics

a. Statistical Analysisb. Classical versus Quantum Statisticsc. The Distribution of Molecular Speedsd. The Maxwell-Boltzmann Distributione. Quantum Statisticsf. Applications of Bose-Einstein Statisticsg. Application of Fermi-Dirac Statistics

7. Nuclear Structure and Radioactivity

a. Nuclear constituentsb. Nuclear Sizes and Shapesc. Nuclear Masses and Binding Energiesd. The Nuclear Forcee. Radioactive Decayf. Conservation Laws in Radioactive Decayg. Alpha Decayh. Gamma Decayi. Natural Radioactivity

The Schrödinger Equation

a. Justification of the Schrödinger EquationA fundamental equation required for describing the wave behavior of nonrelativistic particles.

The conditions required for justification:

1. Energy conservation must be satisfied 2. The equation must be contestant with de Broglie hypothesis 3. The equation must be continuous (linear and single valued).

Like other waves, the free particle can be described by the equation

And for time independent case,) (1)Since

(2)

=E

One dimensional Time-independent Schrödinger equation.

In which, U is the potential energy =U(x). For a free particle U(x)=0.E is the particle’s total energy.m is the particle’s mass, x is the position.

Did this equation satisfy the conditions?

b. The Schrödinger Recipei. Using Schrödinger equation with

discontinuous does not affect the continuity of . To describe the different regions of space we have to write different equations.

ii. Solve Schrödinger equation to find for different situations.

iii. By applying boundary conditions, several solutions can be obtained


Comparison between the use of Newton 2nd law of motion and Schrödinger equation

Newton’s law Schrödinger equation

The object approaches a boundary at which it is subjected to 2 different :Forces F(x) Potential energies U(x)

The basic behavior of the particle is found by solving

Newton’s 2nd law of motion Schrödinger equation =E (x)

The position x The wave function (x)

Of the object is always continuous across the boundary

The velocity The derivative

Is also continuous as long as the

force Change in potential energy

Remains finite

c. Probabilities and Normalization– The wave function describes the wave properties of the

particle.– The squared absolute amplitude of gives the probability

for finding the particle at a given location in space.• The probability density P(x), the probability per unit

length in one dimension, is defined as

The probability to find the particle in the interval dx which lies between x and x+dx.The probability to locate the particle varies smoothly and continuously.


• The probability of finding the particle between x1 and x2 is the sum of all probabilities of finding the particle in the intervals dx, and is given by

• Normalization:Since the total probability over the whole region is 1,

=1Which is called normalization condition.

• Any solution to the Schrödinger equation, for which becomes infinite, must be eliminated.

For example, if the solution is

for the region x>0, then the first term must be eliminated as it leads to infinite value of when x approaches infinity. This is done by using A=0.In the region x<0, the 2nd term must be eliminated by using B=0.

• Average value of x, xav

Any physical quantity depending on the particle’s position will be determined with uncertainty as we are not uncertain about the particle's position itself.The probability of finding the particle at a particular position gives probable outcome of any single physical measurement or average value of a large number of measurements.If x1 is measured a certain number of times n1 and x2 is measured n2 of times, and so on.

d. Applications• The Free ParticleF(x)=0 and so U(x)=constant, doesn’t change for all values of x. Let U(x)=0 and substitute in Schrödinger equation.


The solution of this equation is in the form

which gives the energy values.

• Particle in a Box (one dimension)Consider one dimensional box of length =L, in which a particle is moving freely, with boundaries, U(x)=0 at 0 ≤ x ≤ L, andU(x)=∞ at x<0, x>LThis box is called infinite potential well.Ψ=0 outsideand inside the box. A and B have to be found first.

• At the boundaries, ψ must be continuous, therefore,

at x=0 and at x<0 ψ =0This yields B must be 0Therefore the solution is limited to

at x=L and x>L, ψ =0Therefore, A sin kL=0

Or where n=1, 2, 3,….

In this way we can determine the values of energies the particle can have

xX=LX=0

U=0U=∞ U=∞

The constant A is still undetermined, we can make use of the normalization condition for the whole region inside the box,=1→ =1

The solution is

∫ sin2𝑎𝑥𝑑𝑥=𝑥2−

sin 2𝑎𝑥4 𝑎

Example 5.2

An electron is trapped in a one-dimensional region of length 1X10-10 m. a. how much energy must be supplied to excite the

electron from the ground state to the first excited state? (111 eV)

b. In the ground state, what is the probability of finding the electron in the region from x=0.09X10-10 m to 0.11X10-10 m. (0.38%)

c. in the first excited state, find the probability of finding the electron between x=0 and x=0.25X10-10 m. (0.25)

Example 5.3

• Solution

∫ sin2𝑎𝑥𝑑𝑥=𝑥2−sin

2𝑎𝑥4𝑎

𝐸𝑛=ℏ2𝑛2 𝜋2

2𝑚𝐿2𝑃=∫𝑥1

𝑥2

|𝚿(𝒙 )|𝟐𝑑𝑥

• An electron is trapped in an infinitely deep potential well of width L = 106 fm. Calculate the wavelength of photon emitted from the transition E4 → E3. (472 nm)

• 3.10 The state of a free particle is described by the following wave function

ψ(x) = 0 for x < −3a= c for − 3a < x < a= 0 for x > a

a. Determine c using the normalization condition. (1/2√a)

b. Find the probability of finding the particle in the interval [0, a]. (1/4)

• Show that the average value of x is L/2, independent of the quantum state.

Example 5.3

𝑥𝑎𝑣=∫−∞

+∞

|Ψ (𝑥)|2 𝑥𝑑𝑥

e. The Simple Harmonic Oscillator


f. Time Dependence


g. Steps and Barriers


The Rutherford-Bohr Model of the Atom

a. Basic Properties of Atoms

b. The Thomson Model


c. The Rutherford Nuclear Atom


d. Line Spectra


e. The Bohr Model


f. The Franck-Hertz Experiment


a. The Schrödinger Equation in Spherical Coordinates

3.The Hydrogen Atom in Wave Mechanics

b. The Hydrogen Atom Wave Functions


c. Radial Probability Densities


d. Angular Momentum and Probability Densities


e. Intrinsic Spin


f. Energy Levels and Spectroscopic Notation


g. The Zeeman Effect


a. The Pauli Exclusion Principle

4.Many-Electron Atoms

b. Electronic States in Many-Electron Atoms


c. The Periodic Table


d. Properties of the Elements


e. X-Rays


f. Optical Spectra


g. Lasers


a. The Hydrogen Molecule Ion

5.Molecular Structure

b. The H2 Molecule an the Covalent Bond


c. Other Covalent Bonding Molecules


d. Ionic Bonding


e. Molecular Vibrations


f. Molecular Rotations


g. Molecular Spectra


a. Statistical Analysis

6.Statistical Physics

b. Classical versus Quantum Statistics


c. The Distribution of Molecular Speeds


d. The Maxwell-Boltzmann Distribution


e. Quantum Statistics


f. Applications of Bose-Einstein Statistics


g. Application of Fermi-Dirac Statistics


a. Nuclear constituents

7.Nuclear Structure and Radioactivity

b. Nuclear Sizes and Shapes


c. Nuclear Masses and Binding Energies


d. The Nuclear Force


e. Radioactive Decay


f. Conservation Laws in Radioactive Decay


g. Alpha Decay


h. Gamma Decay


i. Natural Radioactivity


phys 342 modern physics 2. course elements 1.the schrödinger equationthe schrödinger equation...

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