## phy294_quantum.pdf-How likely an event is toshowing page 1-4 out of 12

##### Page 1

How

likely

an

event

is

to

occur

|Ψ|

2

= ΨΨ

∗

just to contrast: use of amplitude^2 to find max intensity in classical

mechanics

Probability of finding a particle over a region

∫

|Ψ|

2

+

3: [−

ℏ

2

∇

2

2

+ (̅)] Ψ(̅, ) = (ℏ)

Ψ(̅, )

1: −

ℏ

2

2

∂

2

Ψ(, )

2

+ ()Ψ(, ) = (ℏ)

Ψ(, )

Partial differential equation that describes the quantum state of a

particle, in term of its position and time.

Exist in free space, where

() = 0

in Schrodinger equation.

Ψ(, ) =

((−)

= ℏ, = ℏ

##### Page 2

Time independent Schrodinger Equation assumes that wavefunctions

can form

standing waves

, known as

stationary states.

The probability of

finding an electron at a position does not depend on time. By this, it

assumes that the wave equation is

separable

into

spatial

part and

temporal

part.

Assume separability

:

Ψ(, ) = ()()

Spatial part

:

−

ℏ

2

2

1

()

∂

2

()

2

+ () = Constant

Temporal part

:

(ℏ)

1

()

ϕ()

= Constant

Solving temporal part :

() = exp{−(/ℏ)}

Time independent

:

|Ψ|

2

=

∗

∙

∗

= ||

2

Additionally, require that

∫

|ψ|

2

= 1

, convergence to 0 as x ->

infinity,

and

to be continuous at boundary

≤ 0, ≥

:

=∞

0<<

:

=0

−

ℏ

2

2

∂

2

()

2

= ()

≤ 0, ≥

:

() = 0

0<<

:

() = √2/

sin , = √2

/ℏ

For

continuity

(0) = 0, () = 0

:

=

,

=

2

2

ℏ

2

/2

2

##### Page 3

≤ 0, ≥

:

=

0

0<<

:

=0

0<<

:

−

ℏ

2

2

∂

2

()

2

= ()

() = sin + cos

= √2

/ℏ

≤ 0, ≥

:

−

ℏ

2

2

∂

2

()

2

= ( − ())()

() = exp(+) [not possible] + exp(−)

= √2(

0

− )/ℏ

2

Continuity:

2 cot =

−

If

>

0

, then sinusoidal waves throughout, lower energy

compared to 0 < x < L region, longer wavelength

If

<

0

, then the wave decays out due to negative kinetic energy

(note: also imaginary momentum)

0

also dictates the number of possible states in a well

−

ℏ

2

2

∂

2

()

2

+

1

2

2

() = ()

Solution:

= ( +

1

2

) ℏ

0

,

0

= √/

, n = 0,1,2…

=(

2

!√

)

1

2

() exp(−

1

2

2

2

)

0

=(

√

)

1

2

exp(−

1

2

2

2

)

,

1

=(

2√

)

1

2

(2) exp(−

1

2

2

2

)

##### Page 4

A number extracted from an operator, which is a mathematical

description of how observables are extracted from wavefunction that

contains all information

Average in distribution:

̅ = ∫ |Ψ()|

2

In general, measurable quantity,

=∫Ψ

∗

Ψ

, operators:

o

Momentum

̂ = −ℏ

o

Position

=

o

Energy

= ℏ

∆ = √∫( − ̅)

2

|Ψ|

2

=

√

2

− ̅

2

Uncertainty principle states that

∆∆ ≥ ℏ/2

∆∆ = ℏ∆∆ = ℏ∆()∆ = (ℏ∆)(∆) = ∆∆ ≥ ℏ/2

The uncertainty of a well-defined observable is zero, and the eigenvalue

of the eigenfunction below has to be a constant.

() = ()