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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: [−
2
2
2
+ (̅)] Ψ(̅, ) = (ℏ)
Ψ(̅, )

1: −
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
:
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
2
2
2
()

2
= ()
 ≤ 0,  ≥ 
:
() = 0
0<<
:
() = √2/
sin  ,  = √2
/ℏ
For
continuity
(0) = 0, () = 0
:
 = 
,
=
2
2
2
/2
2


Page 3
 ≤ 0,  ≥ 
:
=
0
0<<
:
=0
0<<
:
2
2
2
()

2
= ()
() =  sin  +  cos 
 = √2
/ℏ
 ≤ 0,  ≥ 
:
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
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.
() = ()


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