Chapter 2Intro to quantum mechanics

Why do we need QM?

Classical physics is great for large objects, but fails when it comes to atomic particles like electrons and EM waves.

QM picks up where classical mechanics fails.

Semiconductor materials -> properties governed by behavior of electrons in crystal lattice.

QM and more specifically wave mechanics can explain the behavior of electrons in semiconductor crystals.

Schrodinger’s wave equation

Ultraviolet catastrophe

Till 1800 all light problems could be solved by treating light as waves.

The UV catastrophe is an experiment that wave theory couldn’t explain.

To understand the UV catastrophe we need to understand thermal radiation.

You could say...the UV catastrophe was the hottest problem a century ago. Bad pun!

Why do hot objects glow?

Radiation from hot objects

What did classical physics say about thermal radiation?

So as you go to shorter wavelengths the intensity really takes off. (rayleigh and jeans)

According to the Rayleigh and Jeans law at UV the intensity approaches ∞ ! And thus this was called the UV catastrophe.

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I ∝1

λ4

Max Planck

Came up with an equation that mimicked thermal radiation but could not explain how it worked.

Planck introduced the concept of quantization to explain thermal radiation.

E = hν(nu). (h = 6.625 x 10-34 J-s known as the plack’s constant).

If atomic oscillations are the source of the spectra then the energy of each of these oscillators is given by the above equation.

E = hν is also called a packet of energy or a quanta.

Max planck (1858-1947)Nobel prize in 1918 for work on quantum theory

Planck’s radiation law as a function of frequency and wavelength

Photoelectric effect

Fig 2.1

Monochromatic light incident on a material, under certain conditions leads to the ejection of electrons called photoelectrons.

Classical physics says light is a wave – Below threshold intensity – no

emission Above threshold intensity – emission

that increases with intensity – this effect has nothing to do with the frequency of the light.

Photo

ele

ctro

n e

nerg

y

Incident intensity

ITHRESHOLD

PHOTOELECTRIC EFFECT(cont.)

What is observed is that “At a constant incident intensity, max KE of the photoelectron varies linearly with frequency, with a limiting threshold frequency of ν= νo”

IR = 1000 nmRed = 800 nmGreen = 510 nmBlue = 480 nm

RKE-Max IRKE-Max = 0GKE-MaxBKE-Max

BKE-Max>GKE-Max>RKE-Max>IKE-MaxR for same incident intensity!

PHOTOELECTRIC EFFECT(cont.)

If intensity is constant and the frequency is changed then the MAX KE of the photoelectron changes linearly with the frequency.

If the frequency is constant and the Intensity is the changed, the rate at which the photoelectrons emit is changed but the MAX KE is not affected.

Max K

E f

or

a p

hoto

e-

Frequency

νTHRESHOLD

I constant

Einstein’s interpretation of the photoelectric effect in 1905

Light can be treated as a particle.

From planck’s theory we can say that light is a particle or a packet of energy with E = hν.

This packet is called a photon.

Higher the frequency greater the energy. So the energy of a blue photon is greater than than of a red photon.

Explaining the photoelectric effect with photons.

A minimum energy is required to dislodge the photoelectron from the material. This is Emin=hνMin=work function.

Energy in excess of the wf goes into the KE of the electron.

As the frequency gets higher there is more energy directed to the KE.

KEMAX = (1/2)mv2= hν-hνMin

Compton effect

Photons behave like billiard/pool balls – energy and momentum are conserved. More proof of particle behavior.

De Broglie's theory

If waves can be treated as particles can particles be treated as waves?

Principle of wave-particle duality

What is the De-broglie wavelength of an abrams tank? Hint: it weighs 65 tons and travels at a speed of 42 mph.

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p =h

λ

λ =h

p

Davisson and germer – wave nature of electrons.

Read text.

Uncertainty principle

It is impossible to describe with absolute accuracy the position and momentum of a particle.

It is impossible to describe with absolute accuracy the energy and the instant at which the particle had that energy.

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Δp.Δx ≥ h

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ΔE .Δt ≥ h

h =h

2π=1.054 ×10−34 J − s

1st statement

2nd statement

This is only reasonable for sub-atomic particles. Proof: speeding ticket.

Probability density function

If you cant specify where an electron is, then the alternative is to state the likelihood of finding an electron at a particular place i.e probability.

We can also develop probability density functions to determine that an electron has a particular energy.

Schrodinger’s wave equation

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Schrodinger's equation :

−h2

2m∇ 2Ψ(x,y,z, t) +V (x,y,z)Ψ(x,y,z, t) = ih

∂Ψ

∂twhere,

Ψ is the wave function and V is the potential (energy)

Wave eq. (cont)

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This is a partial differential equation

∇ 2 =∇_

.∇_

∇_

=∂

∂xa_

x+∂

∂ya_

y+∂

∂za_

z

∇ 2 =∂ 2

∂x 2a_

x+∂ 2

∂y 2a_

y+∂ 2

∂z2a_

z

Equation 2.1 can be re - written as :

−h2

2m(∂ 2Ψ

∂x 2+∂ 2Ψ

∂y 2+∂ 2Ψ

∂z2) +VΨ = ih

∂Ψ

∂t−(2.2)

Writing equation 2 in 1 - D :

−h2

2m(∂ 2Ψ(x, t)

∂x 2) +VΨ(x, t) = ih

∂Ψ(x, t)

∂t−(2.3)

What does the Ψ mean?

The wave function Ψ(x,t) “describes” the behaviour of a particle of mass m “acted upon” by the potential V.

Ψmay be complex

This is new physics. It cannot be derived from other more basic laws. It can be rationalized for simple cases, such as the free particle (V=0), which is represented by a plane wave.

The “truth” of QM can be judged only by its ability to predict results which agree with observation.