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Phys 4141 – Introduction to Quantum Mechanics

Syllabus – instructor: Prof. Jonathan P. Dowling

(his) office hour: T&Th 12:00-1:00PM

Textbook: Griffths (2nd ed), Feagin (1st Ed)

Already Homework 1

Any question -> to Jon at next week

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What is “Quantum?” cf) classical

Small size (microscopic scale)

How small? Usually comparable size of the Atom

Cf) size of H-atom: Bohr’s radius

( )

( )

Why not classical mechanics? Experimental data require

another type of mechanics. => Quantum Mechanics

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1.1 Schrödinger Equation

A system which consists of a particle with mass in

potential ( ) (for simplicity 1-D)

Classical Mechanics: position ( )

Newton’s 2nd law:

: velocity

: acceleration

: force

: Kinetic energy

Quantum Mechanics : wave function ( )

Schrodinger equation:

: complex number ( )

( : Planck’s constant)

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1.2 Statistical Interpretation

Born’s statistical interpretation

∫ | ( )|

= “probability” of finding the particle

between and , at time

[ | ( )| ( ) ( ) with complex conjugate

( ) ]

Probability gives “indeterminacy.”

Question: When we measure the particle’s position, we

got the result at . Where is the particle just before

the measurement?

1. the realist position (Einstein, d’Espagnat): the particle

was at measured position.

2. the orthodox position (Copenhagen interpretation):

particle wasn’t really anywhere.

3. the agnostic position : refuse to answer

Experiment will decide which is right position.

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Jordan’s comment: “observation not only disturb what is

to be measured, they produce it … We compel (the

particle) to assume a definite position.”

Copenhagen interpretation ( the most widely accepted)

by Bell’s inequality (experimentable)

Question) What if I made a second (same) measurement,

immediately after the first? Same result.

(interpreted as the collapse of wavefunction)

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1.3 Probability

1.3.1 Discrete variables

( ) ( )

with ∑ ( )

Average value of

⟨ ⟩ ∑ ( )

∑ ( )

⟨ ( )⟩ ∑ ( ) ( )

∑ ( ) ( )

Deviation from the average: ⟨ ⟩

⟨ ⟩ ∑( ⟨ ⟩) ( )

∑ ( )

⟨ ⟩∑ ( )

⟨ ⟩ ⟨ ⟩

Variance: ⟨( ) ⟩

Standard deviation : √⟨( ) ⟩ : the measure of the

spread about ⟨ ⟩

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⟨( ) ⟩ ∑( ) ( ) ∑( ⟨ ⟩) ( )

∑( ⟨ ⟩ ⟨ ⟩ ) ( )

∑ ( ) ⟨ ⟩⟨ ⟩ ⟨ ⟩ ⟨ ⟩ ⟨ ⟩

Or √⟨ ⟩ ⟨ ⟩

1.3.2 continuous variables

Probability ( ) vs. probability density ( ( ))

∫ ( )

∫ ( )

: normalization

∫ ( )

⟨ ⟩

∫ ( ) ( )

⟨ ( )⟩

⟨( ) ⟩ ⟨ ⟩ ⟨ ⟩

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Problem 1-3) ( ) ( )

Normalization : ∫ ( )

√

[∫

]

∫ ∫

|

⟨ ⟩ ∫ ( )

∫ (

) ( ) ∫ ( )

2nd moment: ⟨ ⟩ ∫ ( )

∫ (

) ( ) ∫ ( ) ( )

∫ ( )

( ) ∫ ( )

( )√

Standard deviation : √⟨ ⟩ ⟨ ⟩ √

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1.4 normalization

∫ | ( )|

Presence of particles anywhere and anytime:

normalizable ( square-integrable)

Does ( ) stay normalized as time goes on if it is

normalized at ?

Or,

∫ | ( )|

∫ | ( )|

∫

| ( )|

∫ (

)

&

| ( )|

[

(

)]

∫ | ( )|

(

)|

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since ( ) will be vanished as . (or, particle

exists somewhere)

partial integration

∫

∫

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1.5 Momentum

⟨ ⟩ ∫ | ( )|

How to connect this average ⟨ ⟩ with experimental data?

⟨ ⟩ : the average of measurements performed on

particles all in the state ( by the collapse of

wavefunction).

Repeated measurements on same ensemble

the expectation value ⟨ ⟩ is the average of repeated

measurement on an ensemble of identically prepared

system

no velocity concept like in classical Mechanics since

is not defined in

in orthodox position.

Assume now that ⟨ ⟩ ⟨ ⟩

, then :

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time-evolution : ⟨ ⟩

∫ | | ∫

| |

∫

[

(

)]

∫ (

)

∫

momentum: ⟨ ⟩ ⟨ ⟩

∫

( ⟨ ⟩)

Kinetic energy :

(

)

Angular momentum (order of operator)

⟨ ( )⟩ ∫ (

)

Example) For classical quantity , what is

corresponding operator?

( )

( )

Not clear.

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Ehrenfest’s theorem : expectation value obeys the

classical laws.

⟨ ⟩

∫

∫(

)

∫(

)

∫((

)

(

))

∫

⟨

⟩

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1.6 The Uncertainty principle

Where is the wave?

Sinusoidal wave : wavelength is well-defined, but not

position.

Bump (packet): position is well-defined, but not

de Broglie formula,

since (

)

Heisenberg’s uncertainty principle

precise measurements on each position and momentum

measurement, not simultaneous.