5.
73 Lecture #5 5-1
Lecture #5: Continuum Normalization
Last time: Free Wavepacket
encoding of x0, ∆x, p0, ∆p
* use of the Gaussian functional form, G(x; x0, ∆x), to avoid
calculating integrals
* use of stationary phase to encode x0 in |g(k)|eiα(k)
* use g(k) because it is automatic to put in e − iE kt !
For moving and spreading free wavepacket:
∆x is time dependent
∆p is not (because free wavepacket is not subject to any
force)
Today: Normalization of eigenfunctions which belong to continuously (as opposed to
discretely) variable eigenvalues.
• convenience of ortho-normal basis sets: generalization
for continua
• we often talk about “density of states”, but in order to
do that we need to define what we mean by “state”
• computation of absolute probabilities — cannot depend
on how we choose to define “state”.
1. Identities for δ-functions.
2. ψ�k, ψ�p,ψ�E for eigenfunctions that correspond to continuously variable
eigenvalues.
3. finite box with countable number of discrete states taken to the limit L → ∞.
Normalization independent quantity:
⎛ #states ⎞⎛ #particles ⎞
P(x, θ) = ⎜ ⎟⎜ ⎟
⎝ δθ ⎠⎝ δx ⎠
θ is the argument of the delta-function. So if we integrate over a region of θ
and x, we have the absolute probability, ∫∫ d�dx P(x,�).
4. two examples — “predissociation” rate and smoothly varying spectral density.
revised 8/13/20 8:20 AM
5.73 Lecture #5 5-2
In Quantum Mechanics, there are two very different classes of systems.
* SPATIALLY CONFINED: • E is quantized
what is ρE • can count states, easy to compute
good for? density of states dn = ρ E
dE
∞
• can normalize to 1 = ∫ ψ *E ψ Edx
T: classical period of oscillation −∞
1
* # of encounters/sec:
T
L/ v
(v, classical velocity, is dependent on x)
* fraction of time in region of length L:
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&
∞ δij = 0 i≠j orthogonal
δ ij = ∫
*
ψ i ψ j dx
−∞
δij = 1 i=j normalized
ψ has dimension of L–1/2
δij has dimension of pure number. (Kronecker-δ)
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 definitions of a δ-function:
1
δ(x − x′) = δ(x, x′) =
2π ∫ e− iu( x− x′) du.
What is it good for?
shifts a function evaluated at x to
∫ δ(x,x ′)ψ(x)dx=ψ(x ′). the same function evaluated at x′.
δ ( x, x′ ) has dimension of 1/x. (Dirac-δ function)
revised 8/13/20 8:20 AM
5.73 Lecture #5 5-3
Some useful δ-function identities:
We do this so that we will be able to transform between δk, δp, and δE
(where E = f(k)) delta-function normalization schemes.
1
e.g., δ ( p − p ′ ) = δ ( ! ( k − k ′ )) = δ ( k − k ′ )
1
1. δ(ax,ax′) = δ(x,x′)
a !
dimension of p–1
dimension of 1/k
nonlecture proof of #1 above
1
δ(ax,ax′) = −iu ( ax −ax ′)
2π ∫
e du change variables
v = au
dv = a du
1 1 1
δ ( ax,ax′ ) = ∫ e ( ) dv = δ ( x, x′ )
− iv 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|
−1
dg(xi )
2. δ ( g(x) ) = ∑ δ(x, xi ) provided that
!
i dx dg(xi )
≠0
zeros
dx
of g(x)
expand g(x) in the region near each 0 of g(x),
dg
i.e., x near x i g(x) ≅ (x − xi ).
dx x= x i
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 8/13/20 8:20 AM
5.73 Lecture #5 5-4
g(x) = (x – a)(x – b) has zeroes at x = a and x = b.
dg d ⎡ 2
= x − x(a + b) + ab ⎤⎦ = 2x − (a + b)
dx dx ⎣
dg dg
= a −b = b−a
dx x=a
dx x=b
−1
dg(x1 )
δ (g(x)) = ∑
i dx
( )
δ x,xi (zeroes of g(x))
1
= ⎡δ (x,a) + δ (x,b)⎤
a −b ⎣ ⎦
Other examples:
(
δ x2 − a2 = ) 1
⎡δ (x,a) + δ (x − a)⎤
2a ⎣ ⎦
δ (x 1/2
)
− a1./2 = 2a1/2δ (x − a) (a > 0)
See Merzbacher, Quantum Mechanics, 3rd Edition, pages 630-632.
revised 8/13/20 8:20 AM
5.73 Lecture #5 5-5
EXAMPLES
A. g(x) = (x–a)(x–b) This has zeroes at x = a, and x = b.
1
You should show that δ(g(x)) = [δ (x,a) + δ(x, b) ].
a−b
B. δ ( E 1/2 , E ′1/2 )
g (E ) = E 1/2 − E ′1/2 has one zero at E = E ′, expand g(E) about E = E ′, thus for E near E ′
1
g (E ) ± E ′ −1/2 (E − E ′ ).
2
you should show that δ ( E 1/2 , E ′1/2 ) = 2 E ′1/2 δ(E , E ′ )
1/2
⎛ m ⎞
This is useful because k ∝ E1/2 δ(E − E ′ ) = ⎜ 2 [δ(k − k ′ ) + δ(k + k ′ )] for a free particle
⎝ 2h ( E ′ −V0 ) ⎟⎠
1/2
⎛ 2h 2 ⎞
δ ( k E (x ) − k E ′ (x )) = ⎜ (E ′ −V (x ))1/2 δ(E − E ′ )
⎝ m ⎟⎠
or
d
Another property of δ-functions: δ(x , x ′ )
dx
δ(x , x ′ ) is an even function:
d
∴ expect δ(x , x ′ ) ≡ δ′(x , x ′ ) to be an odd function:
dx
d
This is useful because application of δ(x , x ′ ) to f (x ) is capable of picking
dx
df
out evaluated at x′.
dx
Non-lecture:
Use definition of derivative to prove that
∞
∫
−∞
δ′(x, x ′ ) f (x)dx = − f ′( x ′ )
d ⎡δ(x + ε, x ′ ) − δ(x, x ′ ) ⎤⎦
δ(x, x ′ ) = lim ⎣
dx ε→0 ε
∫ δ(x + ε, x ′ ) f (x)dx = f ( x ′ − ε)
∫ δ(x, x ′ ) f (x)dx = f ( x′)
⎡δ(x + ε, x ′ ) − δ(x, x ′ ) ⎤⎦ f ( x ′ − ε) − f ( x ′ )
∴ ∫ lim ⎣ f (x)dx = lim = − f ′( x ′)
ε→0 ε ε→0 ε
revised 8/13/20 8:20 AM
5.73 Lecture #5 5-6
There are several useful ways to normalize wavefunctions.
Bound states. Particle is confined in space (with tunneling
tails outside the box). Space normalized. 1 particle
(mostly) in box.
Box normalized ψ L,E i (x) (box of length L)
∫
*
dx ψ L, E i ψ L, E j = δ ij Kronecker - delta
−∞
OK for bound states, but not continua.
dimension of ψ L,E is L-1/2
dimension of δ ij or δ E iE j is 1.
Continua. We need some other form of normalization.
dx (e ikx ) e ik ′x = ∫
L /2 L /2
∫
* ( )
e.g. dxe − i k −k ′ x
− L /2 − L /2
if k = k ′ we get L
if k ≠ k ′ we expect to get 0 (in limit L → ∞)
So we can normalize to a delta function in E, p, or k.
revised 8/13/20 8:20 AM
5.73 Lecture #5 5-7
∞
∫
*
δE : –∞ dxψ δE,E ψ δE,E
i j
(
≡ δ Ei − E j )
δ(Ei – Ej) has the useful δ-function property:
∫ dE δ ( E − E ) ψ j δE ,E = ψ δE ,E j
This implies that 𝛿(E – Ej) has the dimension of 1/E
and that 𝜓𝛿E,E has dimension of L–1/2E–1/2
p E (x) = [2m( E − V(x))]
1/2
δp : PE2 2 m = E −V (x )
( )
∞
∫–∞ dxψ *δp ,p E ψ δp,p E = δ p E i (x) − p E j (x)
i j
( )
δ p − p ′ has dimension of 1/p
ψ δp,p has dimension of L-1/2 p −1/2
E
δk :
1/2
⎡ 2m ⎤ ! 2 k E2
k E ( x ) = ⎢ 2 ( E −V ( x ) ) ⎥ = E −V ( x )
⎣! ⎦ 2m
( )
∞
∫–∞ dxψ *δk ,k E ψ δk,k E = δ k E i (x) − k E j (x)
i j
( )
δ k − k ′ has units of 1/k
ψ δk,k has units of L-1/2k −1/2
revised 8/13/20 8:20 AM
5.73 Lecture #5 5-8
What are all of these normalization schemes good for?
When you make a measurement on a continuum (unbound)
system, you ask
What is the probability of finding a particle between
x, x + dx
and 𝛳, 𝛳 + d𝛳 ? 𝛳 can be E, pE(x), or kE(x)
The probability is P(x, 𝛳)dxd𝛳
Want P(x, 𝛳). Has dimensions L–1 𝛳–1 (as shown for ψ δ,δE ,ψ δ p ,pE , and ψ δk ,kE )
P(x,θ) = ψ δθ,θ (x)ψ δθ,θ (x)
*
!
There is another less abstract way to get this kind of
information. “Discretize the continuum” by adding an infinite
barrier at x = L and taking the limit L→∞. This way we can use
box-normalized states, and actually count the states.
The WKB quantization condition (will be derived in Lecture #7)
gives
dn (2m)1/2 x + ( E)
∫x ( E) dx( E − V(x))
−1/ 2
=
dE h −
revised 8/13/20 8:20 AM
5.73 Lecture #5 5-9
V(x)
x
x0 L
We have a complicated V(x) for x < x0 and constant for x > x0.
In the region where V(x) is constant at V(x0) = V0.
L
∫x 0
[E − V(x)] −1/ 2 dx = [E − V0 ]−1/2 ( L − x0 ) ∝ L
1
and box normalization causes | ψ |2 ∝
L
⎛ dn ⎞ *
so we get P(x, E) = lim ⎜⎝ L ⎟⎠ ψ L,E (x)ψ L,E (x)
L→∞ dE dimension
dimension
L–1E–1 dimension L–1
E–1
2 Schematic Examples
* Bound → free transition probabilities
* Constant spectral density across a dissociation or ionization
limit.
revised 8/13/20 8:20 AM
5.73 Lecture #5 5 - 10
Bound-Free Transition (predissociation)
bound (box normalized
discrete energy levels)
V(x) E
→L
repulsive (continuum of E-
levels,
can’t really box normalize)
xstationary x
phase
At t = 0, system is prepared in Ψ(x,0) = ψbound(x)
Fermi’s Golden Rule:
2π 2
! ∫
Rate = Γ bound→free = ψ free *
δE ( E )Ĥψ bound
L , E dx ρδE (E )
dn (E ) derive this key quantity by box normalizing
ρδE = δE ⎛ ⎞
dE repulsive state and taking lim ⎜ 1 dn L ⎟
L→ ∞⎝ L dE ⎠
Then compute the Ĥ integral using two box normalized functions.
Constant spectral density on both sides of a bound/free limit
II(𝜔)
(ω)
ω çto bound levels
(ω)
𝜔
çto continuum
Intensity(ω)
~ smooth function of ω, no
ΔE
discontinuity at onset of continuum
v=0
revised 8/13/20 8:20 AM
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https://ocw.mit.edu/
5.73 Quantum Mechanics I
Fall 2018
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