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- Lecture Physics A2: Fundamental of Quantum Mechanics - PhD. Pham Tan Thi
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- Fundamental of Quantum Mechanics
Pham Tan Thi, Ph.D.
Department of Biomedical Engineering
Faculty of Applied Sciences
Ho Chi Minh University of Technology
- Quantum Mechanics
- Diffraction of Light/Electron by One Slit
- Double-slits Experiments for Electrons
(Wave-like character of electrons)
- Wave-Particle Duality
Energy in Energy out
A beam of light can be thought as ....
... a flux of particle ... an electromagnetic wave
(Newton/Planck/Einstein) (Huygens/Maxwell/Hertz)
Electric Field
Wavelength λ
Zero mass, speed: c = 3 x 108 m/s
Direction of propagation
Energy carried by each particle:
h = 6.6262 x 10-34 J.s (Planck) c = λν
Dispersion relation
ν (frequency) = 1/T (period)
(Free space)
Particle (Photon): Wave (Electromagnetism):
- Photoelectric Effect - Interference
- Compton Effect - Diffraction
- de Broglie’s Hypothesis
“Light can behave like a particle and Matter/Electron
can behaves like a wave
Energy of Photon by Special Relativity:
E = pc
Energy of Photon by Planck’s theory:
E = h⌫
de Broglie’s wavelength:
h
=
p
Wave if λ > Scale
It turns out that everything’s kind of mixed together at the fundamental
microscopic scale.
- For a Non-relativistic Free Particle
Momentum is p = mv, here is v is the speed of the particle
Total energy of a free particle, E, is kinetic energy:
h h h
B = = =p
p mv 2mE
p2 mv 2
E=K= =
2m 2
Bullet: m = 0.1 kg; v = 1000 m/s
36
➔ B ⇡ 6.63 ⇥ 10 m
- What is the de Broglie’s Wavelength of an Electron?
What is the de Broglie’s wavelength of an electron moving at 2.2 x 106 m/s
- What is the de Broglie’s Wavelength of an Electron?
What is the de Broglie’s wavelength of an electron moving at 2.2 x 106 m/s
Now this is really really fast: 2.2 million meters per second. But it’s not
relativistic. It’s is still slow compared to the speed of light so we can still
do everything fairly classically.
31
p = mv = 9.1 ⇥ 10 ⇥ 2.2 ⇥ 106 [kg m/s] = 2 ⇥ 10 24
N.s
34 24 10
= h/p = 6.626 ⇥ 10 /2 ⇥ 10 = 3.33 ⇥ 10 m
This is important number. This speed is a kind of average speed of an
electron in ground state of hydrogen.
- Accelerated
What is the de
Charges
Broglie’s
(electrons,
Wavelength
protons)
of a Person?
Produce
de Broglie’s theory can be extended to show that all matter exhibits the
same wave-particle duality as light. This means everything in the universe
can act like a wave:
h
= and h = 6.626 x 10-34 m2kg s-1
mv
This shows that an object’s wavelength gets smaller when the more
massive it is, and the faster it is moving.
If a person has a mass of 75 kg, and is jogging at 8 km/h (which is about
2.2 m/s), then
- Accelerated
What is the de
Charges
Broglie’s
(electrons,
Wavelength
protons)
of a Person?
Produce
de Broglie’s theory can be extended to show that all matter exhibits the
same wave-particle duality as light. This means everything in the universe
can act like a wave:
h
= and h = 6.626 x 10-34 m2kg s-1
mv
This shows that an object’s wavelength gets smaller when the more
massive it is, and the faster it is moving.
If a person has a mass of 75 kg, and is jogging at 8 km/h (which is about
2.2 m/s), then
6.626 ⇥ 10 31 36
= = 4.016 ⇥ 10 m
7.5 ⇥ 2.2
This is about 700 billion billion times smaller than the classical electron
radius, which is about 2.8 x 10-15 m.
Diffraction works best if the slit is about the same size as the
wavelength, and so this explains why we do not notice wave-like behavior
in human.
- Energy in Energy out
Terminology
Particle: Wave:
A beam of light can be thought as ....
Our traditional Our traditional
understanding of a particle understanding of a wave
... a flux of particle ... an electromagnetic wave
(Newton/Planck/Einstein) (Huygens/Maxwell/Hertz)
Electric Field
Wavelength λ
Direction of propagation
“Localized” - “De-localized” -
definite position, spread out in space
momentum, and time
*Disturbance in the
medium
- perposition of these waves has a momentum equal to the average of the two individua
lues of momentum. The amplitude varies, giving the total wave a lumpy character not
ssessed by either individualSuperposition
wave. Principle
Ey(x)
(a) 0 x
Ey(x)
(b) 0 x
- Heisenberg Uncertainty Principle
Heisenberg uncertainty states that only one of the “position” or
“momentum” can be measured accurately at a single moment within the
instrument limit.
or
It is impossible to measure both the position and momentum
simultaneously with unlimited accuracy.
x ! uncertainty in position
px ! uncertainty in momentum
then ~ h
x px ~=
2 2⇡
- Heisenberg Uncertainty Principle
If Δx is measured accurately, i.e., x!0 px ! 1
The principle applies to all canonically conjugate pairs of quantities in
which measurement of one quantity affects the capacity to measure the
other.
For examples, Energy E and time t
~
E t
2
and Angular momentum L and angular position θ
~
L ✓
2
- Determination of the Position of a Particle by a Microscope
Suppose we want to determine accurately the position and momentum
of an electron along x-axis using an ideal microscope free from all
mechanical and optical defects.
The limit of resolution of the microscope is
x=
2sini
here i is semi-vertex angle of the cone
of rays entering the objective lens of
the microscope.
Δx is the order of uncertainty in the x-
component of the position of the
electron.
- Determination of the Position of a Particle by a Microscope
We can’t measure the momentum of the electron prior to illumination. So
there is uncertainty in the measurement of momentum of the electron.
The scattered photon can enter the microscope anywhere between the
angular range +i/-i.
The momentum of the scattered photon is (according to de Broglie)
h
p=
Its x-component can be given as
2h
px = sini
- Determination of the Position of a Particle by a Microscope
The product of the uncertainties in the x-components of position and
momentum for the electron is
✓ ◆
2h
x px = ⇥ sini
2sini
~
x px = h >
2
This is in agreement with the uncertainty relation.
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