Post on 02-Jan-2016
Physics Department
Phys 3650Quantum Mechanics – I
Lecture NotesDr. Ibrahim Elsayed
Quantum
Mechanics
Mechanics based on Newton’s law works fine for macro-particles
The aim of Newtonian mechanics is to find the evolution of a particle position , x(t)
From x(t), we can know everything about the particle
How can we do that, simply if we know the force acting on the particles, F
Then we apply Newton’s 2nd law F = m a
With the help of boundary condition, the velocity, v(t) and position, x(t) can be found
Newtonian mechanics
If the particle velocity is too high v(t), approach speed of light
If the particle mass is too small, like atoms and electrons
Newtonian mechanics Fails
In the first, we use relativistic mechanics
In the second we use quantum mechanics
Newtonian mechanics
Light behaves as wave when it undergoes interference, diffraction etc.
Light wave is completely described by Maxwell's equations
But the wave nature of electromagnetic radiation failed to describe phenomenalike blackbody radiation, photoelectric effect and such
Einstein proposed his idea of photon (quantization of light in quanta, hu)
In this way, Einstein describe the particle-like nature of light.
The beginning of Story
𝐸=h𝜈𝑎𝑛𝑑𝑝=𝐸𝐶
=h𝜆
The associated momentum with a photon of frequency u:
Electrons are known as particles with certain mass
Electrons diffraction Experiment shows that electron has a wave-like nature
de Broglie made a hypothesis that just as radiation has particle-like properties, electrons and other material particles possess wave-like properties
The beginning of Story
𝜈=𝐸h
𝑎𝑛𝑑 𝑝=h𝜆
For free particles,
Electron in hydrogen-like atom moved in circular orbit, The centripetal force equal to the attraction force between the electron and nucleus
Old Quantum Mechanics
�⃗�=𝑟 × �⃗�=𝑚𝑣𝑟=𝑛ℏ ,𝑛=1 ,2 ,3 ,….
The “angular momentum” is quantized.
𝑓 =𝑚𝑎=𝑚 𝑚2
𝑟=4𝜋 𝜀2 𝑍𝑒 .𝑒
𝑟2 +r
Old Quantum Mechanics
𝐸=−𝑚𝑒4 𝑍2
8 𝜀2h21𝑛2
=− 𝑘𝑛2=−2 .181 𝑥10− 18 1
𝑛2𝐽𝑜𝑢𝑙𝑒𝑛=1 ,2,3 ,….
The “angular momentum” is quantized.
, Bohr radius
+r
Energy = Kinetic + Potential
𝐸=1/2𝑚𝑣2+ 𝑍 𝑒2
4𝜋 𝜀21𝑟
The wave length of the emitted lights
Old Quantum Mechanics
Δ 𝐸=−2 .181𝑥10−18( 1𝑛𝑢❑ −
1
𝑛2 ) 𝐽𝑜𝑢𝑙𝑒𝑛=1 ,2 ,3 ,….
nU
nL
EU
EL
Δ 𝐸=2 .181 𝑥10− 18( 1𝑛❑2 −
1
𝑛2 ) 𝐽𝑜𝑢𝑙𝑒𝑛=1 ,2 ,3 ,….
1𝜆
=Δ 𝐸h𝑐
=108 ,680 ( 1𝑛❑2 −
1
𝑛2 )𝑐𝑚−1𝑛=1 ,2 ,3 ,….
The Bohr Theory of the atom (“Old” Quantum Mechanics) worksperfectly for H (as well as He+, Li2+, etc.).
Old Quantum Mechanics
The only problem with the Bohr Theory is that it fails as soonas you try to use it on an atom as “complex” as helium.
Postulate of Quantum Mechanics
Max Born extended the matter waves proposed by De Broglie, by assigning a mathematical function, Ψ(r,t), called the wavefunction to every “material” particle
Ψ(r,t) is what is “waving”
1- Wave function
But how a wave represents a particle?
Localization is the nature of particles (where is the particle: at point (2,1) )
Spread is the nature of wave (where is the wave: every where)
(2,1)
What is the wave length? (It is 0.5 meter)
Where is the jerk? (It is moving over there)
What is wave length of the jerk? (it is not a wave)
If you want to precisely define the position, the less the wavelength is defined
If you want to precisely define the wavelength, the less the position is defined
There is an intermediate case in which:
the wave is fairly well localized and wavelength is fairly well defined
1- Wave function …….
If the particle has a momentum p, the associated wavelength is
Thus the spread in wavelength corresponds to a spread in momentum
2- Uncertainty Principle
𝜆=h𝑝
The best one can do according to Heisenberg Uncertainty principle is:
An experiment cannot simultaneously determine a component of the momentum of a particle (px) and the exact value of the corresponding coordinate, (x).
(∆𝑥 )(∆𝑝 )≥ℏ2
2- Uncertainty Principle ……
Example: Bullet with p = mv = 0.1 kg × 1000 m/s =
100 kg·m/s If Δp = 0.01% p = 0.01 kg·m/s
m 1005.1m/skg 0.01
s J1005.1 3234
px
Which is much more smaller than size of the atoms the bullet made of!So for practical purposes we can know the position of the bullet precisely
2- Uncertainty Principle ……
Example: Electron (m = 9.11×10-31 kg) with energy
4.9 eV Assume Δp = 0.01% p
Which is much larger than the size of the atom!So uncertainty plays a key role on atomic scale
A10m10
102.11005.1
kg·m/s 101.2 0.01%
s/mkg 102.1J106.19.4kg101.922
4628
34
28-
241931
-px
pp
mEp
3-Porobability Density
The probability P(r,t)dV to find a particle associated with the wavefunction Ψ(r,t) within a small volume dV around a point in space with coordinate r at some instant t is called “Probability Density”
dv
r
x
y
z
dV
For one dimension:
𝑃 (𝑥 ,𝑡 ) 𝑑𝑥=|Ψ (𝑥 ,𝑡)|2𝑑𝑥
where
The probability of finding a particle somewhere in a volume V of space isSince the probability to find particle anywhere in space is 1, we have condition of normalization
For one-dimensional case, the probability of finding the particle in the arbitrary interval a ≤ x ≤ b is
∫𝑎𝑙𝑙 𝑠𝑝𝑎𝑐𝑒
❑
|Ψ (𝑟 ,𝑡 )|2𝑑𝑉 =1
𝑃𝑉=∫𝑉
❑
𝑃 (𝑟 ,𝑡 ) 𝑑𝑉=∫𝑉
❑
|Ψ (𝑟 ,𝑡)|2𝑑𝑉
𝑃𝑎𝑏=∫𝑎
𝑏
|Ψ (𝑥 ,𝑡 )|2𝑑𝑥 𝑃𝑒𝑣𝑒𝑟𝑦 𝑤 h𝑒𝑟𝑒=∫−∞
+∞
|Ψ (𝑥 ,𝑡)|2𝑑𝑥
3-Porobability Density …..
If we have such equation:
4-Operators
�̂� 𝑓 =𝑎 𝑓
where an operator acting on the function f gives the same function f multiplying by a factor a.
In this case we call f as the eigen function of the operator with eigen value a.
For example,
the eigen function 𝑑𝑑𝑥
𝑒𝑘𝑥=𝑘𝑒𝑘𝑥
where
In quantum mechanics,
4-Operators …..
Observable quantities like position x, momentum, p
�̂� ( 𝑓 +h )= �̂� 𝑓 + �̂� h
Linear operator satisfy the condition:
all observable quantities are operators
All operators are linear
Operator Linear ?
x2
log
sin
2
2
dxd
Yes
No
No
No
Yes
Yes
√❑
𝑑𝑑𝑥𝑑2
𝑑𝑥2
If I measure the momentum p, what will I get is the expectation value of p,
5-Expectation Value
Expectation value of an observable is its mean value
A class room has 10 students1 get 10/102 get 8/102 get 7/104 get 6/101 get 5/10
What is the average grade of the whole class?
The average grade of the whole class
المثال هذا :فى1هى 10/10إحتمالية
2هى 8/10وإحتمالية 2هى 7/10وإحتمالية 4هى 6/10وإحتمالية 1هى 5/10وإحتمالية
⟨ 𝑥 ⟩=𝑥1𝑝1+𝑥2𝑝2+𝑥3𝑝3+𝑥4𝑝4+𝑥5𝑝5+. . .. . . .. .
𝑝1+𝑝2+𝑝3+𝑝4+𝑝5+. . . .. . . ..=∑ 𝑥 𝑖𝑝𝑖
∑ 𝑝𝑖
5-Expectation Value ……
In the integration form:
Then
⟨ 𝑥 ⟩=𝑥1𝑝1+𝑥2𝑝2+𝑥3𝑝3+𝑥4𝑝4+𝑥5𝑝5+. . .. . . .. .
𝑝1+𝑝2+𝑝3+𝑝4+𝑝5+. . . .. . . ..=∑ 𝑥 𝑖𝑝𝑖
∑ 𝑝𝑖
⟨ 𝑥 ⟩=∫𝑥𝑝 ( 𝑥 ) 𝑑𝑥
∫𝑝 (𝑥 ) 𝑑𝑥
Since
The average (or expectation) value of an observable with the operator  is given by
⟨ 𝑎 ⟩=∫Ψ∗ ( 𝑥 ) �̂�Ψ ( 𝑥 ) 𝑑𝑥
∫Ψ ∗ ( 𝑥 )Ψ (𝑥 ) 𝑑𝑥=∫Ψ∗ (𝑥 ) �̂�Ψ ( 𝑥 ) 𝑑𝑥
Quantum Mechanics
The methods of Quantum Mechanics consist in finding the wavefunction associated with a particle or a system
Once we know this wavefunction we know “everything” about the system!