1. Ch3 Introduction to quantum mechanics
De Broglie waves
Davission-Germer experiment
Uncertainty principles
Wave function
Barrier penetration
Hydrogen atom
Lasers

2. Some words & terms Wave particle duality Wavelength
Electron diffraction
De Broglie wave
Quantum
Uncertainty principle
Wave function
Schrodinger equation
Photon, electron
Sinusoidal, Exponential
Probability density
Potential well
Barrier penetration (势垒穿透)
tunnel
Scanning tunnel microscope (STM)
Operator
Eigenfuction, Eigenvalue
Laser
Spontaneous (simulated) emission

3. Wave-particle duality of light
The dual wave-particle nature of light is now a basic part of the theory of light and matter.
Wave feature: υ, λ;
Particle feature: E, p;
Photons
energy: E=hυ;
momentum: p=h/λ, k=2pi/λ;
k: wave number;

4. De Broglie waves
In1924, Louis de Broglie proposed that matter possesses wave as well as particle characteristics, receiving the Nobel Prize in 1929.
A moving body behaves in certain ways as though it has a wave nature.
Photon wavelength:
De Broglie wavelength:
His doctoral thesis in 1924 contained his proposal that moving bodies have wave properties that complement their particle properties. This feature was confirmed in diffraction experiments with electron beams in 1927.

5. Which property is more important?
Problem: Find the de Broglie wavelengths of a 46g golf ball with a speed of 30m/s and an electron with a speed of 107m/s.
For a 46g golf ball,
For an electron,
Radius of the hydrogen atom, 5.3x10-11m.
Which set of properties is more conspicuous depends on how its de Broglie wavelength compares with its dimension and the dimensions of whatever it interacts with.

6. The diffraction of electrons
In 1927, Davisson and Germer showed that electron beams are diffracted when they are scattered by the regular atomic arrays of crystals.
A beam of electrons from a heated filament (灯丝) is accelerated through a potential difference V. Passing through a small aperture (孔), the beam strikes a single crystal of nickel (镍). Electrons are scattered in all directions by atoms of the crystal.
In 1926, at Bell Telephone laboratories, they were investing the reflection of electron beam from the surface of nickel crystals.

8. The Bragg equation
Instead of a continuous variation of scattered electron intensity with angle, distinct maximum and minimum were observed with their positions depending on the electron energy.
The Bragg equation for maxima in the diffraction pattern is:
In a particular case, a beam of 54eV electrons were directed perpendicularly at the nickel target and a sharp maximum in the electron distribution occurred at an angle of 50o with the original beam. Here θ=650, d=0.091nm, n=1, λ=0.166nm. 54eV0.166nm
Classical physics predicts that the scattered electrons will emerge in all directions with only a moderate dependence of their intensity upon scattering angle and even less upon the energy of the primary electrons.

9. Electron microscopes
The wave nature of moving electrons is the basis of the electron microscope, the first of which was built in 1932.
Fast electrons have wavelengths very much shorter than those of visible light. For example, an electron with 54eV (4.4x106m/s) has the wavelength of 0.166nm.
In an electron microscope, current-carrying coils produce magnetic fields that act as lenses to focus an electron beam on a specimen. A magnification of over 1,000,000 can be reached.
Electrons are easily controlled by electric field and magnetic fields because of their charge.
The technology of magnetic lenses does not permit the full theoretical resolution of electron waves to be realised in practice. For instance, 100keV electrons have wavelengths of nm, but the actual resolution may be only 0.1nm.

10. The uncertainty principle
The principle describes a natural limit to the precision of simultaneous measurements of the position and momentum of particles.
If an object is said be at position of x with an uncertainty of Δx, then any simultaneous measurement of the x component of momentum must have an uncertainty Δpx consistent with
It also implies that either wave or particle properties, but not both, may be observed in a given experiment.

11. The uncertainty relations
Heisenberg uncertainty relation for position and momentum.
It can be used to estimate the size and the energy of the hydrogen atom in its ground state.
Heisenberg uncertainty relation for time and energy.
It is used to study the lifetime of unstable systems.

12. Wave function
Each particle is represented by a wave function, and the square of its absolute magnitude |ψ|2 evaluated at a particular place at a particular time is proportional to the probability of finding the body there at that time. Probability density.
The probability density |ψ|2 is given by the product of ψ and its complex conjugate ψ*.
The wave function must be single-valued, superposable, with finite partial derivatives.
Its absolute squared amplitude gives the probability for finding the particle at a given point in space.

13. Schrodinger equation
The wave function is governed by Schrodinger equation, which is a type of equation known as eigenvalue equation. Solutions can only be obtained for certain values of energy known as energy eigenvalues.
The Schrodinger equation is wave equation in terms of wave function which yields the probability of physical variables in terms of expectation values.

14. Particle in a box
Infinite potential well, finite potential well,
The solution to Schrodinger equation for a particle trapped in a box is just a series of standing de Broglie waves.

15. Barrier penetration (tunneling)
In quantum mechanics, particles can sometimes be found in or pass through regions (E<V) that are forbidden by energy conservation. This is called barrier penetration.
Wave function in this barrier potential takes the form:
Sinusoidal oscillation in the region x<0;
Exponentials in the region 0<=x<=a,
Sinusoidal oscillation in the region x>a;
The particle has a nonzero probability of being found in a region where it can not go according to classical physics. a V

16. According to classical physics, a particle of energy E less than the height U0 of a barrier could not penetrate - the region inside the barrier is classically forbidden. But the wavefunction associated with a free particle must be continuous at the barrier and will show an exponential decay inside the barrier. The wavefunction must also be continuous on the far side of the barrier, so there is a finite probability that the particle will tunnel through the barrier.

17. Scanning tunneling microscope
It is a remarkable application of tunneling.
The instrument mechanically tracks the electronic wave functions outside the surface of a material, producing an extremely accurate representation of the atoms at the surface.
When a thin metal probe is brought very close to the surface of the material. The gap between two surfaces forms the barrier. The electrons from the metal tunnel through the barrier into the probe and creates a current.
The electrons of the surface atoms tunnel through the air space between platinum or tungsten needle tip and the sample surface. In 1986, Gerd Binnig and Heinrich Rohrer were awarded the Nobel Prize for this sensitive instrument.

18. The probe moves horizontally along the surface to maintain the same current. The map of vertical positions of the probe forms the surface structure.
probe
material E

19. Hydrogen atom (I)
In a hydrogen atom, the electron’s motion is restricted by the inverse square electric field of nucleus. The electron is free to move in three dimensions.
Since the electrical potential is a function of r rather than of x,y,z, Schrődinger’s equation is expressed in terms of spherical polar coordinates r,θ,φ.
r: the length of radius vector from origin O to point P;
θ: zenith angle, angle between radius vector and +z axis,
φ: azimuth angle, angle between the projection of the radius vector in xy plane and the +x axis,
zenith angle: 天顶角，
azimuth angle：方位角

20. Hydrogen atom (II)
The wave function can be divided into three different functions which may be governed by three independent equations.
There are three quantum numbers:
Magnetic quantum number m: the direction of the angular momentum,
Orbital quantum number l: the magnitude of the electron’s angular momentum,
Principle quantum number n: the total energy of the electron,

21. How atomic states are denoted?
It is customary to specify electron angular momentum states by a letter.
Sharp, principal, diffuse, fundamental,…
Atomic states are denoted by a combination of the principal quantum number with the letter representing orbital angular momentum, such as 2s (n=2, l=0), 4d(n=4,l=2). l= 1 2 3 4 5 6 …. s p d f g h i

22. Electron probability density
The probability of finding the electron in a hydrogen atom at a distance between r and r+dr from the nucleus is:
For different states, P(r) changes with r differently.
Zenithal probability density has angular preference except an s state (l=0, m=0). Azimuthal probability density is a constant.
Thus, the electron’s probability density is symmetrical about the z axis. It can be called electron cloud.
P(r): radial probability density,

23. Lasers
Laser is an acronym for light amplification by simulated emission of radiation.
Three ways of radiation interacting with atomic energies:
Spontaneous emission, high E to lower E;
Induced absorption, lower E to high E,
Induced (stimulated) emission: A passing photon with the right energy induces the atom to emit a photon and makes a transition to the lower state, producing two photons in phase.
Population inversion is essential for a laser, that is, a higher state has a greater population than a lower state.
Laser: light amplification by stimulated emission of radiation.
原子系统和光的相互作用方式：自发发射、受激吸收、手机发射。

24. He-Ne laser A mixture of He and Ne is in the ratio of 5:1 to 10:1.
The purpose of the He atoms is to help achieve a population inversion.
An electrical current excites He to 20.61eV,
Excited He atoms colliding with Ne atoms to make them to reach 20.66eV, a metastable state,
The laser transition in Ne from the metastable state at 20.66eV to an excited state at 18.70eV, with the emission of a 632.8nm photon.
KK223

25. The high-voltage discharge across the gaseous mixture excited electrons in helium atoms to the 20.61eV state.

26. Usage of lasers
There are many different types of lasers, such as the ruby laser, the carbon dioxide laser etc.
As lasers produce coherent monochromatic light that can be confined to an intense narrow beam.
They are used to replay music in CD players, to weld parts, to measure distance accurately and to study molecular structure. It is also used in medical treatment and surgery.