Understanding Quantum Mechanics: Electrons in Small Spaces and Wave-Like Properties, Slides of Microwave Engineering and Acoustics

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BASICS OF QUANTUM MECHANICS

Interesting things happen when

electrons are confined to small

regions of space (few nm). For

one thing, they can behave as if

they are in an artificial atom.

They emit light of particular

frequencies … we can make a

solid state laser!

GaInP/AInP Quantum Well Laser Diode

S- G expt

(spin)

Single slit

(position)

In a quantum-mechanical system, the measurement we may

be concerned with is position , for which there are

(infinitely) many options, not just two, as in the spin-1/2 S-

G system!

(b)

†Y\

†+\

†-\

!+=°X+†Y\°

2

!-=°X-†Y\°

2

Sz

Quantum Mechanics – kets and operators

The state of electron is represented by a quantity called a state vector or a

ket, , which in general is a function of many variables, including

time.

In PH425, you learned about kets that contained information about a

particle s spin state. We ll be interested in the information contained in

the ket about the particle position , momentum and energy , and how the

ket develops in time.

In PH 425, you learned about the spin operators S 2 , Sz, Sx etc. We ll be

learning about the position , momentum and energy operators.

In PH425, you represented operators as matrices (in different bases), and

kets as column vectors. We will learn to represent operators as

mathematical instructions (for example derivatives), and kets as

functions (wavefunctions).

!

Some terminology and definitions

Each of the operators has a complete set of eigenstates, and

any set can be use to expand the general state.

are the position eigenstates (states of definite position)

is the position operator

are the momentum eigenstates (definite momentum)

is the momentum operator

are the energy eigenstates (definite energy)

is the energy operator

x

p

!

H

p^ ˆ

x^ ˆ

x is a ket that is the eigenstate of position

In the spins course notation, this ket represents a particle that

is located precisely at position x 2

x

2

< ' x

1

< ' x

2

< ' x

N

Reminds you of a delta function, doesn't it?! Well, it should!

Then what is?

! (^) ( x ) = x!

! x

! x = x!

=!

( x )

Then we have the following identifications (not equalities)

!!! (^) ( x )

!!!

( x )

ψ( x ) is NOT a physically accessible quantity; we cannot measure

it in the laboratory. The physically meaningful quantity is |ψ( x )|

2 .

This is the probability density - the probability per unit volume

in 3D (or probability per unit length in 1D) of finding the particle

in an infinitesimally small region located at x.

The probability of finding this particle somewhere in the universe

must be 1. This statement is represented by:

In bra-ket notation:

!( x ) " # * (^) ( x )# (^) ( x ) = # (^) ( x )

2

!( x ) dx

"#

$ =^ %^ *^ (^ x )%^ (^ x ) dx

"#

$ =^1

!! = 1

This suggests that

! dx

"#

$

This the probability density also tells us about the

probability of finding a particle in a certain region of

space, say between x = a and x = b.

Notation alert: Script P with an argument of x is used for

probability density. The same script P with no x

argument is used for probability. They have different

dimensions!

a < x < b

= !( x ) dx

a

b

= # * ( x )# ( x ) dx

a

b

We will state two things without proof, and you'll see why they

are reasonable, later.

1. In the "position representation" or "position basis", the

position operator is represented by the variable x :

1. In the "position representation" or "position basis", the

momentum operator is represented by the derivative with

respect to x :

1. This follows if you accept (2). The energy operator is:

Now think about eigenfunctions of these operators (worksheet)

p^ ˆ =˙! i!

d

dx

x^ ˆ =˙ x

ˆ H =

p ˆ

2

2 m

ˆ V =˙!

!

2

2 m

d

2

dx

2

+ V ( x )

Look more closely at the momentum eigenfunction or

eigenstate:

1. Why did we change C to p/hbar? And why the subscript?
2. What is the probability distribution for this state?
3. Is it normalized? Normalizable?
4. It is degenerate (new word, maybe?)
5. What sort of particle would be represented by this function?

Position eigenstates:

This is a useful (but a bit pathological) representation of a

position eigenstate:

1. Normalizable?
2. Otherwise reasonable?

! p

( x ) =^ Ae

± ipx /!

! x '

( x ) =^ "^ ( x^ #^ x ')

• Review language of PH
• Kets and wave functions
• Probability density
• Operators – position, momentum, energy
• Eigenfunctions
• Mathematical representations of the above

BASICS OF QUANTUM MECHANICS

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