5.61 Fall 2007
Lecture #8
page 1
QUANTUM MECHANICAL PARTICLE IN A BOX
Summary so far:
∞
∞
V(x)
0
0
n2 h2
En =
8ma 2
a
x
(
)
V (0 ≤ x ≤ a ) = 0
V x < 0, x > a = ∞
k=
(
)
⎛ nπ x ⎞
ψ ( 0 ≤ x ≤ a ) = B sin ⎜
⎝ a ⎟⎠
ψ x < 0, x > a = 0
n
nπ
a
λ=
2a
n
n = 1,2,3,...
()
What is the “wavefunction” ψ x ?
Max Born interpretation:
()
ψ x
2
() ()
=ψ * x ψ x
is a probability distribution or probability density
for the particle
∴
()
ψ x
2
dx is the probability of finding the particle in the interval
between x and x + dx
This is a profound change in the way we view nature!! We can only know the
probability of the result of a measurement – we can’t always know it with
certainty! Makes us re-think what is “deterministic” in nature.
Easy implication:
⇒
∫
x2
x1
()
Normalization of the wavefunction
2
ψ x dx = probability of finding particle in interval
xxx≤≤
12
The total probability of finding the particle somewhere must be 1.
For a single particle in a box,
5.61 Fall 2007
Lecture #8
∫
∫
∞
−∞
a
0
()
2
ψ x dx =
∫ ( )
a
0
2
ψ x dx = 1
⎛ nπ x ⎞
B 2 sin 2 ⎜
dx = 1
⎝ a ⎟⎠
()
ψn x =
⎛ nπ x ⎞
2
sin ⎜
a
⎝ a ⎟⎠
2
⇒
Normalization condition
B=
n = 1,2,3,...
( )
(
page 2
2
a
Normalized wavefunction
()
)
Interpretation of ψ x
ψ 4 = 2 a sin 4π x a
2
( ) (
3
()()
2232sin3axaψπ=
)
( )
(
)
( )
(
)
ψ
2
2
= 2 a sin 3π x a
2
ψ 2 = 2 a sin 2 2π x a
2
ψ 1 = 2 a sin 2 π x a
based on measurement
Each measurement of the
position gives one result.
Many measurements give
a probability distribution
of outcomes.
Expectation values or average values
For a discrete probability distribution
0.4
0.35
0.3
0.25
e.g.
0.2
P
0.15
0.1
0.05
0
2
4
2
6
8
x
10
12
14
5.61 Fall 2007
Lecture #8
page 3
<x > = average value of x
= 4(0.1) + 6(0.1) + 8(0.2) + 10(0.4) + 12(0.2)
= 4(P4) + 6(P6) + 8(P8) + 10(P10) + 12(P12)
where Px is probability that measurement yields value “x ”
x = ∑ xPx
⇒
Now switch to continuous probability distribution
()
Px
→
ψ x
∑
→
∫
x =
⇒
Similarly
x2 =
∫
∞
−∞
∫
∞
−∞
2
dx
()
2
x ψ x dx
()
2
x 2 ψ x dx
Often written in “sandwich” form
() ()
= ∫ ψ * ( x ) x ψ ( x )dx
x =
x2
∫
∞
−∞
ψ * x xψ x dx
∞
2
−∞
For particle in a box
x =
⎛ nπ x ⎞
2 a
x sin 2 ⎜
dx
∫
a 0
⎝ a ⎟⎠
5.61 Fall 2007
Integrate by parts
Lecture #8
→
x =
a
2
The average particle position is in the middle of the box.
page 4