1. Quantum Chemistry / Quantum Mechanics
Postulates

2. Time-dependent Schrödinger Equation
Without potential E = T
With potential E = T + V
Erwin Rudolf Josef Alexander Schrödinger
Austrian
1887 –1961

3. Postulates of Quantum Mechanics
The state of a quantum mechanical system is completely specified by a wave function ψ (r,t) that depends on the coordinates of the particles (r) and time t. These functions are called wave functions or state functions.
For 2 particle system:
Wave function contains all the information about a system.
wave function  classical trajectory
(Quantum mechanics) (Newtonian mechanics)
Meaning of wave function:
P(r) = |ψ|2 =
=> the probability that the particle can be found at a particular point x and a particular time t. (Born’s / Copenhagen interpretation)

4. Implications of Born’s Interpretation
Positivity:
P(r) >= 0
The sign/phase of a wavefunction has no direct physical significance:
The positive and negative regions of this wavefunction both
correspond to the same probability distribution.
(2) Normalization:
i.e. the probability of finding the particle in the universe is 1.

5. 3) Continuity
4) Continuity of first derivatives
5) (Complex)

6. Acceptable or Not ??

7. Postulate 2
To every physical property, observable in classical mechanics, there corresponds a linear, hermitian operator in quantum mechanics.

8. Hermitian Operator
Hermitian operators have two properties that forms the basis of quantum mechanics
Eigen value of a Hermitian operator are real.
Eigenfunctions of Hermitian operators are orthogonal to each other or can be made orthogonal by taking linear combinations of them.

9. Identifying the operators

10. Hermitian operator

11. Postulate 3
In any measurement of the observable associated with the operator , the only values that will ever be observed are the eigenvalues ‘a’ which satisfy the eigenvalue equation:
This is the postulate that the values of dynamical variables are quantized in quantum mechanics.

12. Postulate 4
For a system in a state described by a normalized wave function , the average or expectation value of the observable corresponding to A is given by:

13. Add 4: Correspondence principle 1913/1920
For every physical quantity one can define an operator. The definition uses formulae from classical physics replacing quantities involved by the corresponding operators
Niels Henrik David Bohr
Danish
QM is then built from classical physics in spite of demonstrating its limits

14. Postulate 5
The wave function of a system evolves in time in accordance with the time dependent Schrödinger equation:

15. Postulates – simplified version
1) Properties of any system are completely by the wavefunction, (r,t).
2) The wavefunction and its derivatives are continuous, finite, unique and in general complex
3) Any measurable/observable quantity is obtained as a mean value
4) |(r,t)|2 is probability
5)  satisfies Schrödinger equation

16. Schrödinger Representation – Schrödinger Equation
Time dependent Schrödinger Equation
Hamiltonian kinetic potential energy energy
Sum of kinetic energy and potential energy.
The potential, V, makes one problem different form another H atom, harmonic oscillator.
Q.M.
one dimension
three dimensions
recall
Copyright – Michael D. Fayer, 2007

17. Getting the Time Independent Schrödinger Equation
Copyright – Michael D. Fayer, 2007

18. Pierre Simon, Marquis de Laplace
Kinetic energy
Classical quantum operator
In 3D :
Calling the laplacian
Pierre Simon, Marquis de Laplace
( )

19. < YI Y> <Y IOI Y > Dirac notations
Chemistry is nothing but an application of Schrödinger Equation (Dirac)
< YI Y> <Y IOI Y >
Dirac notations
Paul Adrien Dirac 1902 – 1984
Dirac’s mother was British and his father was Swiss.

20. Uncertainty principle
the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the momentum of the particle uncertain; and conversely, that measuring the momentum of a particle precisely makes the position uncertain
We already have seen incompatible operators
Werner Heisenberg
German

21. p and x do not commute and are incompatible
For a plane wave, p is known and x is not (Y*Y=A2 everywhere)
Let’s superpose two waves…
this introduces a delocalization for p and may be localize x
At the origin x=0 and at t=0 we want to increase the total amplitude,
so the two waves Y1 and Y2 are taken in phase
At ± Dx/2 we want to impose them out of phase
The position is therefore known for x ± Dx/2
the waves will have wavelengths

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