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Review : Quantum mechanics
Jae-hoon Ji
Nano Electro-Mechanical Device Lab
Understanding of nature by classical mechanics
Newton’s mechanics
Solid mechanics
Governing Equation
Fluid mechanics
P = ( AE ⁄ L) δ ⋅ = k δ
Understanding of nature by classical mechanics
Electromagnetism
Governing equation
Wave & particle duality
However, many experiment results that could not be interpreted by classical model were reported
Waves behave as particles
Particles behaves as waves
– Photoelectric effect
– Compton effects
– Black body radiation
– Diffraction
– Young’s double-slit experiment
New governing equation - Schrödinger equation
Governing equation
The role of Newton's laws and conservation of energy in
classical mechanics
Understanding of Quantum mechanics with the view of EM wave
EM wave
Wave equation
Wave property?
 First, we should know
Dispersion relationship
Dispersion relationship
𝑘 = ω με
Equation for understanding EM wave
QM wave
Wave equation
Ψ : Contains all the
measurable information
about the particle
Dispersion relationship
𝑃 = 𝑘ℎ
𝐸 = ℎ𝑤
Equation for understanding wave
Understanding of Quantum mechanics with the view of EM wave
Cf) Classical harmonic oscillator – Mass on a spring
Undamped oscillator case
Understanding of Quantum mechanics with the view of EM wave
Mathematics for understanding QM
The measurable information is obtained by wave function, governing equation.
By Mathematics (Differential equation)
Unacceptable forms of
It is quite important to consider the form of equation and boundary condition
ψ must be continuous partial derivatives must also be continuous
Understanding of Quantum mechanics with the view of EM wave
Mathematics for understanding QM
1. The eigenfunctions of Hermitian operators are orthogonal
2. Any wavefunction can be expanded with
Cf) Linear algebra
Hamiltonian operator: The operator associated with Energy
3. Coefficients calculated by integration due to
orthogonality
Eigenvalue of this operation -> Energy
Wave function : Contains the measurable information
From the view of mathematics, calculating energy of a wave is nothing but calculating the Eigenvalue of equation.
Understanding of Quantum mechanics with the view of EM wave
Mathematics for understanding QM
Calculation to something you can observe in the
laboratory, the "expectation value" of the measurable
parameter
We can calculate expected value of given wave function.
Ex)
Understanding of Quantum mechanics with the view of EM wave
Understanding of Quantum mechanics with the view of EM wave
EM wave
1D – Perfect conductor wall
QM wave
1D - Infinite potential wall
−1 ∙ 𝑒 𝑖𝑘𝑧
?
1 ∙ 𝑒 𝑖𝑘𝑧
r=-1 (Reflection coefficient)
?
Understanding of Quantum mechanics with the view of EM wave
EM wave
QM wave
1D – Dielectric
1D - finite potential wall
𝑟 ∙ 𝑒 𝑖𝑘𝑧
𝑒 𝑖𝑘𝑧
k1
k2
Wave is continuous
1st derivative of wave is continuous
Ʈ ∙ 𝑒 𝑖𝑘𝑧
Understanding of Quantum mechanics with the view of EM wave
Previous case
QM wave
1D - finite potential wall
Previous case
Previous case
Understanding of Quantum mechanics with the view of EM wave
QM wave
EM wave
1D - Infinite potential well
1. Equation
2
d2

 ( z )  E ( z )
2
2m dz
2. Wave
function
& B.C
2. Wave function
& B.C
1. Equation
E y ( z  0, L)  0
k
d2
2
E
(
z
)

k
z E( z)  0
2
dz

k z   2   k x 2

E  E0 sin(k z z ), (k z 
n
)
L
Understanding of Quantum mechanics with the view of EM wave
EM wave
2. Wave function
& B.C
1. Equation
Understanding of Quantum mechanics with the view of EM wave
QM wave
1. Equation
2. Wave function
& B.C
kL
kL
 B cos
2
2
kL
kL
De  L /2  A sin
 B cos
2
2
  L /2
kL
kL
Ce
  A cos  B sin
k
2
2

kL
kL
 De  L /2  A cos  B sin
k
2
2
Ce  L /2   A sin
𝑤κ 𝑘𝑤
𝑘𝑤
=
tan
2
2
2
𝑤κ 𝑘𝑤
𝑘𝑤
−
=
cot
2
2
2
Understanding of Quantum mechanics with the view of EM wave
L
κ
L
κ
L
κ
L
L
κL
L
κ κ
κ κ
𝑤κ 𝑘𝑤
𝑘𝑤
=
tan
2
2
2
𝑤κ 𝑘𝑤
𝑘𝑤
−
=
cot
2
2
2
Understanding of Quantum mechanics with the view of EM wave
1. Equation
QM wave
How about coupled potential well?
1. Individually
1 2
1
2
As a
matrix
1. Individually
1. Equation
2. Close together
2.a Close look at [S]
2.b Close look at [H]
We can find Eigen state and eigenfunction
Understanding of Quantum mechanics with the view of EM wave
Crystal structure
Bloch wave
Wavefunction for a particle in a periodically-repeating environment,
most commonly an electron in a crystal
Multiply a plane wave by a periodic function
1. Under the periodic potential
2. The eigenstates ψ of the Hamitonian
Isosurface of a Bloch wave
in silicon lattice
Understanding of Quantum mechanics with the view of EM wave
Crystal