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!