Upload
dodieu
View
227
Download
2
Embed Size (px)
Citation preview
STM image of atomic “quantum corral”
[email protected]@2015.3 2
Atoms form a quantum corral to confine the
surface state electrons.
3.1 General Postulates of Quantum
Mechanics
• P1: To every quantum system there is a state function, Ψ(�,�), that contains everything that can be known about the system
• P2:
(a) Every physical observable O (position, momentum, energy, etc.) is associated with a linear Hermitian operator. ��
(b) Eigenvalue problem: ���� = ����
(c) If a system is in the initial state Ψ, measurement of O will yield one of the eigenvalues λn of �� with probability � �� = |�Ψ(�,�)��
∗��� |�
[email protected]@2015.3 3
3.1.2 Eigenvalues and Eigenfunctions
[email protected]@2015.3 5
3.1.3 Hermitian Operator
• Hermitian operators have real eigenvalues. Their
eigenfunctions form an orthogonal, complete set of
functions.
[email protected]@2015.3 6
(if normalized)
3.1.4 Operators for Quantum
Mechanics
• Momentum operator
• Energy operator
[email protected]@2015.3 7
3.1.4 Operators for Quantum
Mechanics
• Position operator
[email protected]@2015.3 8
The eigenfunction is
3.1.4 Operators for Quantum
Mechanics
• Commutation and the Uncertainty principle
[email protected]@2015.3 9
α and β operators are commute
The difference operator: is commutor
�
So one cannot measure x and px
(along x-axis) with arbitrary
precision
They are not commute!
3.1.4 Operators for Quantum
Mechanics
• Uncertainty principle
[email protected]@2015.3 10
So one can measure x and py
(along y-axis) with arbitrary
precision
3.1.5 Measurement Probability
• Postulate 3: The mean value of an observable
is the expectation value of the corresponding
operator.
• Postulate 4:
[email protected]@2015.3 11
3.2.1 Boundary Conditions on
Wavefunction
[email protected]@2015.3 13
Consider a one-dimensional space with
electrons constrained in 0<x<L
Evidence for existence of electron wave
[email protected]@2015.3 14
3.3 Analogies between Quantum
Mechanics and Classical Electromagnetics
• Maxwell’s equations:
[email protected]@2015.3 15
comparison
3.4 Probabilistic current density
[email protected]@2015.3 16
3.5 Multiple Particle Systems
• State function
• Joint probability of finding particle 1 in d3r1
point r1
and finding particle 2 in d3r2
of point
r2
• State function obeys
[email protected]@2015.3 17
3.5 Multiple Particle Systems
• Hamiltonian:
• Example: two charged particles:
[email protected]@2015.3 18
3.6 Spin and Angular Momentum
• Lorentz force
• If the particle has a net magnetic moment µ, passing through a magnetic field B
• Angular momentum:
• Spin is a purely quantum phenomenon that cannot be understood by appealing to everyday experience. (it is not rotating by its own axis.)
[email protected]@2015.3 19
3.7 Main Points
• Meaning of state function
• Probability of finding particles at a given space
• Probability of measuring certain observable
• Operators, eigenvalues and eigenfunctions
• Important quantum operators
• Mean of an observable
• Time-dependent/independents Schrodinger equations
• Probabilistic current density
• Multiple particle systems
[email protected]@2015.3 20