1.  Heisenberg’s Matrix Mechanics Schrödinger’s Wave Mechanics
The Schrödinger Wave Equation
- Each particle in a physical system is describable by a wave function Ψ that is a function of position and time, that is Ψ(x,t).
- In its time-dependent form for a particle of total energy E moving in a potential field V in one dimension:
(2-23)
in three dimensions
(2-24)

2. Basic Postulates of Quantum Mechanics
||2 or p(x) is called the probability density function. It is measurable and is just the
probability per unit length. P(x) must be normalized:
The wave function (x,t) and its space derivative must be single-valued, finite, and
continuous.
Each physical observable Q is related to a quantum mechanics operator Qop, from
which the expectation value Q can be calculated:

3. separated to obtain an equation containing (t),
Let (x,t) be represented by (x)(t), the Schrödinger equation (2-23) can be
separated to obtain an equation containing (t),
(2-26)
And the time-independent Schrodinger equation:
(2-27)
Total derivatives!
The Potential Well Problem L x
V(x) ∞
V(x) = 0, <x<L
V(x) = ∞, x = 0, L
(2-28)

4. Inside the well we set V(x)=0 in Eq. (2-27)
(2-29)
Possible solution of Eq. (2-29)
Since outside the well =0 and the interior wave must match the exterior wave at the walls
In order that (x) be continuous everywhere,
Then, k must be some integral multiple of π/L
The total energy En for each value of integer n can be solved.
(2-33)

5. The nth wave function
The constant A is found by normalizing n
(2-35)
Energy is quantized; the possible energy
of En are called energy levels
The integer n is called a quantum number
The particular n and corresponding En
describe the quantum state of the particle
n=1 defines the ground state; its energy E1
is not zero
Quantum states defined by n>1 are called
excited states
(t)=e-jt by solving (2-26)stationary state

6. Device application Chap 10.1 Scanning Tunneling Microscope (STM)
Figure 2—6
Quantum mechanical tunneling: (a) potential barrier of height V0 and thickness W; (b) probability density for an electron with energy E < V 0, indicating a non-zero value of the wave function beyond the barrier.
- Even for a particle with E<V0, quantum mechanics predicts that there is a finite
probability the particle will be found beyond the barrier. The mechanism by which
the particle “penetrates” the barrier is called quantum mechanical tunneling.
Zener effect Chap 5.4.
Device application Chap 10.1
Scanning Tunneling Microscope (STM)

7. 2.5 Atomic Structure and The Periodic Table
The Hydrogen Atom
(2-45)
The spherical coordinate system
Quantum Numbers
n principle quantum number n=1, 2, 3, …
l orbital angular momentum quantum number l=0, 1, 2, 3,…, n-1
m magnetic quantum number m= -l, -l+1, …0, 1, …, l-1, l
s spin quantum number
Every allowed energy state of the electron in a H atom is uniquely described by the
above four quantum numbers

8. The Periodic Table—Quantum Mechanics Explanation
—The discovery of an empirical periodic table of elements, 1869, Demitri Mendeleev
—The elucidation of the underlying physical basis of the periodic table, late 1920s
Pauli Exclusion Principle: No two electrons in an atom may have the same set of quantum
numbers (n, l, m, s)
The electrons in an atom tend to occupy the lowest energy levels available to them
Only one electron can be in a state with a given (complete) set of quantum numbers
(Pauli exclusion principle)
—Some conventions:
Principle quantum number n=1, 2, 3, 4, … shells
The naming of l values l=0, 1, 2, 3, 4, …
s, p, d, f, g, …
subshells
Electron state
3p6
Total electronic configuration of Si s22s22p63s23p2