Particles (matter) behave as Particles (matter) behave as waves waves

and the Schrödinger and the Schrödinger EquationEquation1.1. Comments on quiz 9.11 and 9.23.Comments on quiz 9.11 and 9.23.

2.2. Topics in particles behave as waves:Topics in particles behave as waves: The (most powerful) experiment to prove a The (most powerful) experiment to prove a

wave: interference. wave: interference. Properties of matter waves.Properties of matter waves. The free-particle Schrödinger Equation.The free-particle Schrödinger Equation. The Heisenberg Uncertainty Principle.The Heisenberg Uncertainty Principle. The not-unseen observer (self study).The not-unseen observer (self study). The Bohr Model of the hydrogen atom.The Bohr Model of the hydrogen atom.

3.3. The second of the many topics for our The second of the many topics for our class projects.class projects.

4.4. Material and example about how to Material and example about how to prepare and make a presentation (ref. prepare and make a presentation (ref. Prof. Kehoe) Prof. Kehoe)

today

Review: properties of matter Review: properties of matter waves waves

The de Broglie wavelength of a The de Broglie wavelength of a particle:particle:

The frequency:The frequency: The h-bar constant:The h-bar constant: The connection between particle The connection between particle

and wave:and wave:

Wave number and angular Wave number and angular frequency:frequency:

h p

f E h

2k 2 2T f

2h

p k

E

The free-particle Schrödinger The free-particle Schrödinger EquationEquation

The matter waves: The matter waves:

The interpretation of the matter wave function:The interpretation of the matter wave function:

The plane wave solution and verification:The plane wave solution and verification:

22

2

Ψ Ψ

2

x,t x,ti

m x t

2probability density = Ψ x,t

Ψ i kx tx,t Ae

22 2 2 2

2

22 2 2 2

i kx ti kx t i kx t i kx tkAe p

ik Ae Ae Aem x m m m

i kx ti kx t i kx tAe

i i i Ae Aet

2 2 2

22 2

i kx t i kx tp Ae AeE , i

m m x t

Quantum but classical account for energy E, not relativistic

Erwin Schrödinger, 1887-1961, Austrian physicist, shared 1933 Nobel Prize for new formulations of the atomic theory.

Understand this plane waveUnderstand this plane wave

How many of you reviewed the How many of you reviewed the discussions about waves in mechanics?discussions about waves in mechanics?

Complex exponential:Complex exponential:

Probability to find the particle: Probability to find the particle:

Ψ cos sini kx tx,t Ae A kx t iA kx t

ReΨ cos sin 2A kx t A kx t

ImΨ sinA kx t

2probability density = Ψ Ψ Ψ*x,t x,t x,t

2= i kx t i kx tAe Ae A Equal probability to find the particle anywhere the location of this particle is uncertain, although the momentum of this particle is certain. Why?

p k

The Heisenberg Uncertainty The Heisenberg Uncertainty PrinciplePrinciple

Particle-wave duality Particle-wave duality uncertainties uncertainties Plane wave of free electron:Plane wave of free electron:

Momentum Momentum is certain.is certain. Location (where to find the particle)Location (where to find the particle)

is not certain (equal probability).is not certain (equal probability).so so

On the detection screen:On the detection screen: Location is known (measured).Location is known (measured). Momentum (Momentum ( ) is not certain. ) is not certain.

Uncertainty? Review standard deviation:Uncertainty? Review standard deviation:

Ψ i kx tx,t Ae

p k

xp

withi i i i i i

i i i i

p p n p np p

n n

0p x ?

The Heisenberg Uncertainty The Heisenberg Uncertainty PrinciplePrinciple

Because of a particle’s wave nature, it is Because of a particle’s wave nature, it is theoretically impossible to know theoretically impossible to know precisely both its position along on axis precisely both its position along on axis and its momentum component along that and its momentum component along that axis; axis; ΔΔx and cannot be zero x and cannot be zero simultaneously. There is a strict simultaneously. There is a strict theoretical lower limit on their product:theoretical lower limit on their product:

This is called the Heisenberg uncertainty This is called the Heisenberg uncertainty principle (Nobel Prize 1932). principle (Nobel Prize 1932).

Show example 4.4 and 4.5 (student Show example 4.4 and 4.5 (student work).work).

2xp x

Solar system Atom model analogous to the solar system is wrong

Electron waves in an atom

Werner Heisenberg (1901 – 1976), German physicist. Nobel Prize in 1932 for the creation of quantum mechanics.

Example 4.6, an application of Example 4.6, an application of the uncertainty principlethe uncertainty principle

Find the ground state of a Find the ground state of a hydrogen atom (student hydrogen atom (student work).work). The classical mechanics The classical mechanics

approach.approach.

The quantum mechanics The quantum mechanics approach.approach.

The ground state, the minimum The ground state, the minimum mechanical energy for the mechanical energy for the electron.electron.

2

08classical

eE

r

2 2

202 4matter wave

eE

mr r

2 2

3 20

04

dE e

dr mr r

2110

2

45 3 10 m 0 053nmr . .

me

418

2 2 20

2 2 10 J 13 6eV32min

meE . .

minE

most probabler

The energy-Time Uncertainty The energy-Time Uncertainty Principle and the discussions in Principle and the discussions in

section 4.5section 4.5 The energy-time uncertainty Principle:The energy-time uncertainty Principle:

In particle physics, we estimate some particle’s lifetime In particle physics, we estimate some particle’s lifetime by measuring its energy (mass) uncertainty. by measuring its energy (mass) uncertainty. example: the particle example: the particle ππ00 has a mass of 134.98 MeV, has a mass of 134.98 MeV, decays into two photons. Its mean lifetime decays into two photons. Its mean lifetime ττ = 8.4×10 = 8.4×10-17-17 sec, derived from its width of 0.0006 MeV in its mass sec, derived from its width of 0.0006 MeV in its mass measurement. measurement.

By measuring the energy spread (uncertainty) of an By measuring the energy spread (uncertainty) of an emitted photon, we estimate the time an atom stays at a emitted photon, we estimate the time an atom stays at a certain excited state. certain excited state.

The Not-Unseen Observer: please read section 4.5 The Not-Unseen Observer: please read section 4.5 after the class. Discuss with me in office hours if after the class. Discuss with me in office hours if you have questions about this section.you have questions about this section.

2E t

22

17

6 6 10 MeV s

0 0006MeV 11 10 s

E t .

t .

The Bohr Model of the The Bohr Model of the hydrogen atomhydrogen atom

The classical approach:The classical approach:

Bohr postulates: the electron’s angular Bohr postulates: the electron’s angular momentum momentum LL may only take the values: may only take the values:

2 2

20

1

4

e vm

r r

22

0

1

2 8

eKE mv

r

E can take any value, for r is continuous.

where 1 2 3L n n , , ...

Because L mvr mvr n We have

2

0 22

4r n

me

Coulomb force hold the electron in place:

2 2

20

1

4

e vm

r r

For n = 1, the Bohr radius 2

00 2

40 0529nma .

me

2 4

2 2 220 0

1 113 6 eV

8 2 4

e meE .

r n n

The energy

2

08

eE U KE

r

Niels Henrik David Bohr (1885 – 1962), Danish physicist. Nobel Prize in 1922 for work on atomic structure.

The Bohr Model of the The Bohr Model of the hydrogen atomhydrogen atom

2 4

2 2 220 0

1 113 6 eV

8 2 4

e meE .

r n n

Hydrogen spectrum

2197 MeV fm

cE , c

n

7 12 2

1 1 11 0972 10 m

1R R .

n

2n 2n 3n

Review questionsReview questions

If a particle is confined inside a If a particle is confined inside a boundary of finite size, can you be boundary of finite size, can you be certain about the particle’s velocity certain about the particle’s velocity at any given time? at any given time?

Why the Bohr’s hydrogen model is Why the Bohr’s hydrogen model is flawed?flawed?

If you have problem in If you have problem in understanding example 4.6, you understanding example 4.6, you need to see me in my office hour.need to see me in my office hour.

Preview for the next classPreview for the next class Text to be read:Text to be read:

In chapter 5:In chapter 5: Section 5.1Section 5.1 Section 5.2Section 5.2 Section 5.3Section 5.3 Section 5.4Section 5.4

Questions:Questions: How would you generalize the Schrödinger How would you generalize the Schrödinger

equation we have discussed in chapter 4 to equation we have discussed in chapter 4 to include conservative forces?include conservative forces?

Give an example of a classical bound state.Give an example of a classical bound state. If a particle falls into a potential well (a If a particle falls into a potential well (a

quantum well), how do you determine whether quantum well), how do you determine whether it is bounded? it is bounded?

Class project topic, 2 and Class project topic, 2 and how to prepare for a presentationhow to prepare for a presentation

Quantum physics in renewable Quantum physics in renewable energy.energy.

How to prepare for a presentation:How to prepare for a presentation: Reference:Reference:

http://www.physics.smu.edu/kehoe/3305F07/3305Present.pdfhttp://www.physics.smu.edu/kehoe/3305F07/3305Present.pdf

http://www.physics.smu.edu/kehoe/hep/WhyMass.pdfhttp://www.physics.smu.edu/kehoe/hep/WhyMass.pdf

Homework 6, due by 10/2Homework 6, due by 10/2

1.1. Problem 13 on page 134.Problem 13 on page 134.

2.2. Problem 16 on page 134.Problem 16 on page 134.

3.3. Problem 43 on page 137.Problem 43 on page 137.

4.4. Problem 54 on page 138.Problem 54 on page 138.