5.73 Lecture #5
5-1
Continuum Normalization
Last time: Gaussian Wavepackets
How to encode x in
∫
g( k )e ikx dk = ψ(x)
or k in
∫
g( x )e −ikx dx = ψ(k)
stationary phase: good for cooking or inspecting wiggly
functions and for crudely evaluating
integrals of wiggly integrands.
vgroup ≠ vphase
Today: Normalization of eigenfunctions which belong to continuously (as opposed to
discretely) variable eigenvalues.
convenience of ortho-normal basis sets
we often talk about “density of states”, but in order to do
that we need to define “state”
computation of absolute probabilities — cannot depend on
how we choose to define “state”.
1. Identities for δ-functions.
2. ψδk, ψδp,ψδE for eigenfunctions corresponding to continuously variable
eigenvalues.
3. finite box with countable discrete states taken to the limit L → ∞.
Normalization independent quantity:
 # states   # particles 
 δθ  

δx
δθ is the argument of the delta-function. So if we integrate over a
region of θ and x, we have the absolute probability.
4. two examples — “predissociation” rate and smoothly varying spectral
density.
revised 9/11/02 10:58 AM
5.73 Lecture #5
5-2
In Quantum Mechanics, there are two very different classes of systems.
* SPATIALLY CONFINED:
• E quantized
• can count states, easy to compute
density of states dn = ρE
dE
what is ρE
good for?
∞
*
• can normalize to 1 = ∫−∞ ψ E ψ Edx
T: classical period of oscillation
* # of encounters /sec:
1
T
* fraction of time in region of length L:
L /v
T
* SPATIALLY UNCONFINED: • E continuously variable
dn
• can’t count states, so how to compute
?
dE
**
• can ask what is the absolute probability of finding
the system between E, E + dE and x, x + dx
For confined systems, we can express ortho-normalization in terms of Kronecker-δ
δ ij = ∫
∞
ψ *i ψ j dx
−∞
δij = 0
i≠j
orthogonal
δij = 1
i=j
normalized
For unconfined systems, we are going to ortho-normalize states to Dirac δ-functions
In order to do this we need to know better what a δ--function is and what some
of its mathematical properties are.
One of several equivalent definition of δ - function:
1
δ( x − x′) = δ(x, x′) =
e −iu(x − x ′)du.
2π
What is it good for?
∫
∫
δ(x, x′)ψ(x)dx = ψ(x′).
shifts a function evaluated at x to
the same function evaluated at x′.
revised 9/11/02 10:58 AM
5.73 Lecture #5
5-3
Prove some useful Identities
We do this so that we will be able to transform between δk, δp, and δE
(where E = f(k)) normalization schemes.
1.
δ(ax, ax′ ) =
1
δ( x, x′ )
a
e.g., δ (p − p′ ) = δ ( h( k − k′ )) =
1
h
δ ( k − k′ )
nonlecture proof
δ(ax, ax′ ) =
1
e − iu ( ax −ax′ )du
∫
2π
change variables
v = au
dv = a du
1 1
1
e −iv ( x − x ′)dv = δ ( x, x′ )
2π a
a
but, since δ(ax, ax′ ) = δ(ax − ax′) = δ(ax′ − ax) = δ ([−a](x − x′))
1
(δ is an even function), δ(ax,ax′) =
δ( x, x′)
|a|
δ (ax, ax′ ) =
2.
δ g(x) = ∑
i
{
(
)
∫
dg(x i ) −1
δ (x, x i )
dx
provided that
zeros
of g(x)
dg(x i )
≠0
dx
expand g(x) in the region near each 0 of g(x),
i.e., x near x i
g(x) ≅
dg
dx
x =x i
(x − xi ).
If there is only 1 zero, then identity #1 above gives the
required result. It is clear that δ(g(x)) will only be nonzero
when g(x) = 0. Otherwise we need to carry out the sum in
identity #2.
revised 9/11/02 10:58 AM
5.73 Lecture #5
5-4
EXAMPLES
A. g(x) = (x–a)(x–b) This has zeroes at x = a, and x = b.
You should show that δ( g( x)) =
1
[δ(x, a) + δ(x, b) ].
a−b
B. δ(E1/2 , E′1/2 )
has one zero at E = E′, expand g(E) about E = E′, thus for E near E′
g(E) = E1/2 − E′1/2
1
g(E) ±ƒ E′ −1/2 (E − E′).
2
(
)
you should show that δ E1/2 , E′1/2 = 2 E′1/2 δ(E, E′)
m


δ(E − E′) =  2

 2 h (E′ − V0 ) 
1/2
This is useful because k ∝ E
Another property ofδ - functions:
δ( x, x′) is an even function:
1/2
δ( k − k ′)
d
δ( x, x′)
dx
d
δ( x, x′) ≡ δ′(x, x′) to be an odd function:
dx
d
This is useful because
δ( x, x′) is capable of picking
dx
df
out
evaluated at x′.
dx
∴ expect
Non-lecture:
Use definition of derivative to prove that
∞
∫−∞ δ′(x, x′)f(x)dx = −f ′(x ′ )
[δ (x + ε , x ′ ) − δ (x, x ′ )]
d
δ (x, x ′ ) = lim
dx
ε
ε →0
∫ δ (x + ε , x′)f(x)dx = f(x′ − ε )
∫ δ (x, x′)f(x)dx =
∫
∴ lim
ε→0
f(x ′ )
[δ(x + ε, x′) − δ(x, x′) ] f ( x)dx = lim f ( x′ − ε) − f ( x′) = −f ′( x′)
ε
ε→0
ε
revised 9/11/02 10:58 AM
5.73 Lecture #5
δk
δ *
5-5
Our goal is to create ortho-normalized ψ’s that look like eikx:
“normalized to a δ-function in k”
1 ∞ ix(k − k ′ )
∞
δ (k ′, k) = ∫−∞ ψ δ*k,k′ ψ δ k,k dx ≡
dx
∫ e
2π −∞
std. defn. of
δ−function in
k
ψ δk,k ≡ (2π) −1/2 e ikx
∴
for V(x) = constant.
ψ δ k,k is said to be “normalized to δ (k, k ′ )”.
What is the probability of finding the system, which
is described by ψ δ k,k , to be located between 0 ≤ x ≤ L ?
L
0
∫
*
ψ δk,k
ψ δk,k dx =
1 L
L
dx
=
= P δk (L)
∫
0
2π
2π
probability grows without limit as L → ∞
But, more interestingly, what is the probability of finding a system
in a δk-normalized state within a region of length equal to one
de Broglie λ?
λ = h /p =
2π
k
Pδk (λ) =
λ
1
=
2π k
δk normalized states (for V(x) = constant) have: 1/k particle per λ
of ∆x
 or 1 particle per unit length 


2π
revised 9/11/02 10:58 AM
5.73 Lecture #5
5-6
What about ordinary space normalization?
ψ k = N k e ikx
ψ k ′ = N k e ik ′x
2 ∞
i( k ′ − k)x
*
dx
∫ ψ k ψ k ′ dx =|N k | ∫−∞ e
0 if k ≠ k′
∞ if k = k′
⇑
THIS IS THE PROBLEM!
Can’t specify Nk.
GENERALIZE
δ (k, k ′ ) ≡
*
1 ∞ iu(k− k ′ )
∞
e
du = ∫−∞ ψ δ k,k ψ δ k, k ′ dx
∫
−∞
2π
−1/2 ikx
e
where ψ δk,k ≡ (2π)
, thus δ ( k , k ′ ) =
1
2π
∫
∞
−∞
e i( k ′ − k ) x dx
notation δ(k , k ′ ) = δ(k − k ′ ) = δ(k − k ′ , 0)
when δ(k,k′) is multiplied onto f(k) and integrated over all k,
we get f(k′)
∞
∫−∞
δ(k , k ′ )f(k)dk = f( k ′ )
δ(k,k′) is “zero” when k ≠ k′ and is “one” when k = k′
1∂
ψ δk , k 0 is eigenfunction of kˆ =
with eigenvalue k 0 : kˆ ψ δk , k 0 = k 0ψ δk , k 0
i ∂x
ψ δx, x 0 is eigenfunction of xˆ = x
with eigenvalue x 0
revised 9/11/02 10:58 AM
5.73 Lecture #5
5-7
Other normalization schemes for free particle
ψ δp,p = N p e ipx/h
δp
δ
*
what is value of N p ?
[
]
δ(p, p ′ ) = N *p′ N p ∫ exp ix ( p / h − p ′ / h ) dx
= N *p′ N p 2πδ(p / h, p ′ / h)
1442443
−1
1
using
δ(p,p ′ )
identity #1
h
−1/2 ipx/h
∴ ψ δp,p = (2πh)
e
1
h
particle per λ =
p
p
L
L
*
∫0 ψ δ p,p ψ δ p,pdx = 2πh
(
ψ δ±E = N E e ikx ± e −ikx
δδE *
)
 2m( E − V0 ) 
k= 



h2
1
particle per unit * length
h
1/ 2
degenerate pair of
states
you show that
ψ δ+E,E
 m 
=
 2Eπ 2 h 2 
1/4
 2mE  1/2 
cos  2  x 
 h 


ψ δ−E,E
 m 
=
 2Eπ 2 h 2 
1/4
 2mE  1/2 
sin  2  x 
 h 


ψ δ+E, E is orthogonal to ψ δ−E, E
∫
λ
0
+
+*
ψ δE
, E ψ δE, E dx

 1
1
1
from ψ δ−E, E  =
=
+  another
 E
2E
2E 
probability for
δE - normalized
state per λ
* Volume of N-dimensional phase space occupied by a δp normalized state is hN
revised 9/11/02 10:58 AM
5.73 Lecture #5
Thus there are
5-8
1
particles per λ for a δ E - normalized state.
E
1/ 2
or 2 m
particles per unit length
 h 2E

( )
So we have assembled all the basic stuff we will need, at least for
V(x) = constant problems. Now use it to examine a problem we
understand perfectly.
ψEn
 2
= 
 L
1/2
sin   2mE n 

  
2
cos   h 
1/2
n even

x
 n odd
 2mE n 
kn = 

 h2 
-L/2 L/2
1/2
 h2  2
En = 
n
2
 8mL 
1 particle per box of length L

1
particle per unit length

L
 → 0 as L → ∞
λ
particle per λ 
L

4 normalization schemes (δk, δp, δE, box): each gives different #/L or #/λ .
Why - because each scheme defines “state” differently.
# particles   # states  must be independent of
However, expect that 


 normalization scheme
δθ
δx
k, p, E or box
Why? Because the probability of finding a system between x, x + dx
AND θ, θ + dθ is observable. We have completely specified what counts
as an observation.
revised 9/11/02 10:58 AM
5.73 Lecture #5
5-9
Normalization-Independent Quantity for general V(x):
 dn 
dθ3
L→∞ 1
2
 1 L/2
  # states   # particles 
*
 L ∫−L / 2 ψ δθ ψ δθ dx  =  δθ  

δx
lim
density of
states
(# states per
unit θ
The infallible way to get the invariant reference density is to box
normalize (so that one can count states) and then take limit L → ∞.
Why? Because most realistic potentials become smooth and flat at
large enough x.
V(x)
x=L
x_(E) - inner turning point
Procedure:
1. Box normalize ψ E n
(E is quantized)
dn
from E(n)
dE
1 dn
 dn


→ ∞ but
3. take limit L → ∞ 
remains finite


 dE

L dE
2. Compute
example:
2
ψE =  
 L
0
L
∫
L
0
1/2
 2mE  1/2 
n
sin 
x
2 

h


ψ E* n ψ E n dx = 1 by construction (for box normalization)
revised 9/11/02 10:58 AM
5.73 Lecture #5
5 - 10
h2
En = n
8mL2
dE 2nh 2
=
dn 8mL2
2
ρE (E) =
2 1/ 2
8mE n L 
n=
 h 2 
dn 2L m 1/ 2
=
dE
h 2E
( )
REFERENCE DENSITY


1/ 2
 2 m 1/ 2
 dn 1 L
  =  2 m  = P (x, x + δx; E, E + δE )
*

=
dx
lim 
ψ
ψ
E E
 h 2 E 
L→∞  dE L ∫0
δx δE
{144244
3 h  2E 
424
3
1
#
#


indep. of L

 ∆E
L
THIS SECTION TO BE REPLACED
dn δp
dn δk
dn δE
for ψ δE ,
for ψ δp ,
for ψ δk
dE
dp
dk
δE
 2m 
 h 2 E 
±
1/2
= lim
L→∞
above
derived
reference
density
∫
 m 
 2 
 2h E 
∴
δp
dn δE 1 L *
ψ δE ψ δE dx
0
dE 1
L 44
2443
dnδE
= 2( E E ) = 2
dE
1/2
derived earlier
2 because each E is doubly
degenerate
1 L *
1
ψ δ p ψ δ p dx =
∫
0
L
h
dn δ p 1  2m 1/ 2
=
dp h  h 2 E 
∴
2m 1/ 2 2m p
= 
=
=
p
E
dp
 E 
dn δ p
revised 9/11/02 10:58 AM
5.73 Lecture #5
δk
1
L
L
∫0
5 - 11
1
2π
dn δk 2m k
= 2 =
dk
h k E
*
ψ δk
ψ δk dx =
dn δθ dθ
δ E 2E / E
δp p / E
δk k / E
see the pattern?
These results are only valid for
→ ∞ problem. They illustrate all
continuum normalization problems where it is desired to calculate
probabilities.
2 Schematic Examples
* Bound → free transition probabilities
* Constant spectral density across a dissociation or ionization
limit.
revised 9/11/02 10:58 AM
5.73 Lecture #5
5 - 12
Bound-Free Transition (predissociation)
V(x)
bound (box normalized
discrete energy levels)
E
→L
repulsive (continuum of Elevels,
can’t really box normalize)
xstationary
x
phase
at t = 0 system is prepared in Ψ(x,0) = ψbound(x)
Fermi’s Golden Rule:
Rate = Γ bound→free =
n (E)
ρδ E = δ E
dE
2π
ˆ bound dx 2 ρ (E)
ψ δfree*
∫
E (E)Hψ E
δE
h
derive this key quantity by box normalizing
1 dn 
repulsive state and taking lim 
L→∞  L dE 
ˆ integral using two box normalized functions.
Then compute the H
Constant spectral density on both sides of a bound/free limit
I(ω)
ω
( ω)
v = 0
Intensity(ω )
~ smooth function of ω, no
∆E
discontinuity at onset of continuum
revised 9/11/02 10:58 AM