pyl-100-2016-qmlect-01-intro.pdf

8/17/2019 PYL-100-2016-QMLect-01-Intro.pdf

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PYL100 Electromagnetic Waves and Quantum Mechanics

Syllabus

Wave-particle duality, deBroglie waves; Quantummechanical operators; Schrodinger

equation, Wave

function,

Statistical interpretation, Superposition

Principle,

Continuity Equation for probability density; Stationary

states, Bound states, Free-particle solution, 1-D infinitepotential well, Expectation values and uncertainty

relations; 1-D finite potential well, Quantum mechanical

tunneling and alpha decay, Kronig-Penny model and

emergence of bands


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Quantum Mechanics:

Books Quantum Physics of Atoms, Molecules, Solids, Nuclei and

Particles 2 Edition By: Robert Eisberg, Robert Resnick,

Publisher: Wiley (2006), Approx. Rs. 500/-

Introduction to Quantum Mechanics 2 Edition by: David J.

Griffiths, Publisher: Pearson (2005) Approx. Rs. 605

Lecture Materials at

Moodle


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Young’s Double Slit experiment

Maxwell’s theory describes light as electromagnetic waves

S1

S2

I1

I2

I 1+I 2

Reduce

Intensity

Particles

Hit the

Screen ?


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Light as particles

Certain amount of momentum and energy hit the screen at

this point

Momentum: = ℏ = ℎ

Energy: = ℏ = ℏ (2)

Is identical for each particle as long as wavelength is

fixed.

Since = ,E=pc

From relativity, E = cp + m4 These are mass less particles

Shock: Light consists of particles. In order to see them, your light source

has to be extremely weak.


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The problem

S1

S2

How can you have no hits, when both slits

are open?

Particles normally either take path throughS1 or S2.

=> Waves are needed !

Photons were not found this way !

Photons were predicted by Einstein based on fairly complicated

thermodynamical and statistical arguments


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Photoelectric effect

In metals Some electrons are communal.

W, work function: energy required for the electron

to be removed from the metal

K.E.

W

K= Energy given to electron -W

Einstein showed that:

Light consists of photons each carrying

Energy ℏ

High Intensity low

frequency light

Millions of photons

Can not lift the electrons out of atom

K= ℏ -Wℏ

Light


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Compton's Scattering

When light of wavelength λ bounces off a static electron, it is found that its

wavelength gets altered

+

Can be explained from energy and momentum conservation

in a collision between electron and photon

Momentum: = ℏ

Energy: = ℏ

Conclusively proved that light consists of particles, called photons



Wave-particle duality• If light consists of particles called photons Particles may have wave

associated with them.

This is unexpected ?

Why these electrons show an

Interference pattern ? ( in contrast

To bullets)

(Hint: the separation of the minima

Depends on wavelength)

•de Broglie postulated that with every electron of momentum p there is anassociated wave with wavelength : = 2ℏ/

S1

S2

Electron

Array of electron detector



Example of wave nature of electrons

Reflection high energy electron diffraction (RHEED)

pattern of Si(111)7 × 7 surface


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Act of Measurement in Quantum mechanics

S1

S2

Electron

I1+I2

If all electrons are

catched

If 10 % of electrons are

missed

I2

I1

1. An electron acts like it went through one particular slit if we see it doing that

2. The electron acts like it did not have a specific path (through a specific slit) when it

is not seen.


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The wave function• The double slit experiment tells us that electrons do not follow Newtonian

trajectories.

•

Instead their fate is determined by a wave function ψ(x, y, z).

Born's statistical interpretation of the wave function:

Probability is the area under the graph


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Collapse of the wave function

Immediately after measurement

The particle is at point C.

• Quantum Mechanics only offer statistical information.

• Suppose I do measure the position of the particle, and I find it to be at point C.Question: Where was the particle just before I made the measurement?

Collapse of the wave functionThe particle wasn't really anywhere. It is the

act of measurement that forced the particle to

"take a stand" .


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Role of Probability in QM

P r o b a b i l i t y

P (

x )


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Some Postulates of Quatum mechanics

• Postulate 1 The complete information on the state of the particle is

encoded in a complex function ψ(x) called the wave function.

• Postulate 2 The probability of finding the particle between x and x + dx is

given by |ψ(x)|2 dx .

• Postulate 3 If x is measured, ψ will collapse to a spike at the measured

value of x .

• Postulate 4 The evolution of ψ with time is given by the Schrödinger

equation


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PROBABILITY

Question 1. If you selected one individual at random from this group,

what is the probability that this person's age would be 15?

Answer: One chance in 14

=>In general,

one person aged 14,

one person aged 15,

three people aged 16,

two people aged 22,

two people aged 24.

five people aged 25.

N(l4) = 1,

N(15) = 1,

N(16) = 3,

N(22) = 2,

N(24) = 2,

N(25) = 5,

The total number of people in the room is


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PROBABILITY

one person aged 14,

one person aged 15,


two people aged 22,

two people aged 24.


N(l4) = 1,

N(15) = 1,

N(16) = 3,

N(22) = 2,

N(24) = 2,

N(25) = 5,


Question 2. What is the most probable age?

Answer: 25, obviously; five people share this age, whereas at most

three have any other age.

In general, the most probable j is the j for which P(j) is a maximum .


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PROBABILITY

one person aged 14,

one person aged 15,


two people aged 22,

two people aged 24.


N(l4) = 1,

N(15) = 1,

N(16) = 3,

N(22) = 2,

N(24) = 2,

N(25) = 5,


Question 3. What is the median age?

Answer: 23, for 7 people are younger than 23, and 7 are older.

In general, the median is that value of j such that the probability of

getting a larger result is the same as the probability of getting a

smaller result.


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PROBABILITY

one person aged 14,

one person aged 15,


two people aged 22,

two people aged 24.


N(l4) = 1,

N(15) = 1,

N(16) = 3,

N(22) = 2,

N(24) = 2,

N(25) = 5,


Question 4. What is the average (or mean) age?

Answer:

In general, the average value of j

is called expectation value in QM.


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PROBABILITY

one person aged 14,

one person aged 15,


two people aged 22,

two people aged 24.


N(l4) = 1,

N(15) = 1,

N(16) = 3,

N(22) = 2,

N(24) = 2,

N(25) = 5,


Question 5. What is the average of the squares of the ages?

Answer:

In general,

≠


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PROBABILITY

• They have the same median (5), the same average, the same most probable

value, and the same number of elements (10).

• We need a numerical measure of the amount of "'spread" in a distribution,

with respect to the average

is as often negative as positive

and compute the average :

Find out how far each

individual deviates from the

average


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PROBABILITY

We define the variance of the distribution;

standard deviation.

Usually,

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