1. UNIT 1
Quantum Mechanics

2. Schrödinger equation 1926, Erwin Schrödinger (Austria)
Describe a particle with wave function
Wave function has full information about the particle

3. The Born interpretation of the Wave Function
Contains all the dynamic information about the system
Born made analogy with the wave theory of light (square of the amplitude is interpreted as intensity – finding probability of photons)
Probability to find a particle is proportional to
It is OK to have negative values for wave function
Max Born
Probability Density

4. The Born interpretation of the Wave Function

5. The Born interpretation of the Wave Function

6. where i is the square root of -1.
The Schrödinger wave equation in its time-dependent form for a particle of energy E moving in a potential V in one dimension is:
where i is the square root of -1.
The Schrodinger Equation is fundamental equation of Quantum Mechanics.
where V = V(x,t)

7. General Solution of the Schrödinger Wave Equation when V = 0
The solution
This works as long as:
which says that the total energy is the kinetic energy.

8. General Solution of the Schrödinger Wave Equation when V = 0
In free space (with V = 0), the general form of the wave function is
which also describes a wave moving in the x direction. In general the amplitude may also be complex.
The wave function is also not restricted to being real.
Only the physically measurable quantities must be real. These include the probability, momentum and energy.

9. Normalization and Probability
The probability P(x) dx of a particle being between x and x + dx is given in the equation
The probability of the particle being between x1 and x2 is given by
The wave function must also be normalized so that the probability of the particle being somewhere on the x axis is 1.

10. Properties of Valid Wave Functions
Conditions on the wave function:
To avoid infinite probabilities, the wave function must be finite everywhere.
The wave function must be single valued.
The wave function must be differentiable. This means that it and its derivative must be continuous. (An exception to this rule occurs when V is infinite.)
In order to normalize a wave function, it must approach zero as x approaches infinity.

11. Time-Independent Schrödinger Wave Equation
The potential in many cases will not depend explicitly on time.
The dependence on time and position can then be separated in the Schrödinger wave equation. Let:
which yields:
Now divide by the wave function y(x) f(t):
The left side depends only on t, and the right side depends only on x. So each side must be equal to a constant. The time dependent side is:

12. Time-Independent Schrödinger Wave Equation
This equation is known as the time-independent Schrödinger wave equation, and it is as fundamental an equation in quantum mechanics as the time-dependent Schrodinger equation.
So often physicists write simply:
where:
is an operator.

13. Eigenvalues and eigenfucntions
Eigenvalue equation
(Operator)(function) = (constant factor)*(same function)
Operator
Eigenfunction
Eigenvalue
Solution : Wave function
Allowed energy (quantization)

14. Expectation Values
The expectation value, , is the weighted average of a given quantity. In general, the expected value of x is:
If there are an infinite number of possibilities, and x is continuous:
Quantum-mechanically:
And the expectation of some function of x, g(x):

15. Momentum Operator To find the expectation value of p,
With k = p / ħ we have
This yields
This suggests we define the momentum operator as
The expectation value of the momentum is

16. Position and Energy Operators
The position x is its own operator.
Energy operator: The time derivative of the free-particle wave function is
Substituting w = E / ħ yields
The energy operator is:
The expectation value of the energy is:

17. Operators Position Momentum Potential energy Kinetic energy
Total energy

18. Quantum Numbers
Definition: specify the properties of atomic orbitals and the properties of electrons in orbitals
There are four quantum numbers
The first three are results from SchrÖdinger’s Wave Equation

19. Orbital Quantum numbers
An atomic orbital is defined by 3 quantum numbers:
n l ml
Electrons are arranged in shells and subshells of ORBITALS .
n  shell
l  subshell
ml  designates an orbital within a subshell

20. Quantum Numbers Symbol Values Description
n (major) 1, 2, 3, .. Orbital size and
energy = -R(1/n2)
l (angular) 0, 1, 2, .. n-1 Orbital shape or
type (subshell)
ml (magnetic) -l..0..+l Orbital orientation
in space
Total # of orbitals in lth subshell = 2 l + 1

21. Quantum Theory Model Orbitals
One “s” orbital
Three “p” orbitals
Five “d” orbital

22. Ehrenfest’s theorem
Ehrenfest’s theorem states that quantum mechanics provides the same results as classical mechanics for a particle for which the average or expectations values of dynamical quantities are involved.