Chapter-3 Quantum Mechanics of Electronsqli/ECE685/LectureNotes/Chapter-3 Quantum...Chapter 3...

Nanoelectronics

Chapter 3 Quantum Mechanics of

Electrons

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STM image of atomic “quantum corral”

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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(�,�)��

∗��� |�

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3.1 General Postulates of Quantum

Mechanics

• 3.1.1 Operators

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3.1.2 Eigenvalues and Eigenfunctions

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3.1.3 Hermitian Operator

• Hermitian operators have real eigenvalues. Their

eigenfunctions form an orthogonal, complete set of

functions.

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(if normalized)

3.1.4 Operators for Quantum

Mechanics

• Momentum operator

• Energy operator

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3.1.4 Operators for Quantum

Mechanics

• Position operator

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The eigenfunction is

3.1.4 Operators for Quantum

Mechanics

• Commutation and the Uncertainty principle

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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

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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:

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3.2 Time-independent Schrodinger’s

Equation

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Separation of variables

3.2.1 Boundary Conditions on

Wavefunction

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Consider a one-dimensional space with

electrons constrained in 0<x<L

Evidence for existence of electron wave

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3.3 Analogies between Quantum

Mechanics and Classical Electromagnetics

• Maxwell’s equations:

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comparison

3.4 Probabilistic current density

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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

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3.5 Multiple Particle Systems

• Hamiltonian:

• Example: two charged particles:

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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.)

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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

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3.8 Problems

• 1, 3, 8, 9, 15

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