4.5. TIME-DEPENDENT HAMILTONIAN: THE DIRAC PICTURE
109
and the fine structure constant αf is given by
αf =
αf =
e2
1
=
~c
137
(CGS)
e2
1
=
~c4πǫ0
137
(SI).
Exercise: hydrogen-like atom in a magnetic field: Zeeman effect.
The total Hamiltonian for an electron in a magnetic field consists of
H0 =
Hso =
p~ 2
+ V (r),
2me
1 1 dV (r) ~ ~
L · S,
2m2e c2 r dr
and
HB =
~
−e|B|
(Lz + 2Sz ).
2me c
Studying the effect of HB by using the eigenvalues of H0 + Hso , one obtains the
~ field:
Lande’s formula due to the B
1
−e~B
m 1±
.
∆EB =
2me c
(2l + 1)
4.5
Time-dependent Hamiltonian: the Dirac picture
We consider a time-dependent Hamiltonian
H = H0 + V (t),
with
H0 |ni = En(0) |ni .
The time-evolution operator of H is not e−iHt/~ since H = H(t). For evaluating the time
evolution of a quantum state subject to H(t), we introduce the Dirac picture.
4.5.1
Dirac picture / Interaction picture
We define for states:
|α, t0 ; tiI = eiH0 t/~ |α, t0 ; tiS .
110
CHAPTER 4. APPROXIMATION METHODS: PERTURBATION THEORY
At t = 0, the state in the interaction picture coincides with the state in the Schrödinger
picture:
| iI = | iS .
We define for operators:
ÂI ≡ eiH0 t/~ ÂS e−iH0 t/~ .
For the time-dependent potential ÂS = V ,
V̂I = eiH0 t/~ V̂ e−iH0 t/~ .
Recall that the time-evolution of quantum states and operators in the Schrödinger picture
and Heisenberg picture were given by
|αiH = eiHt/~ |α, t0 = 0; tiS
and ÂH = eiHt/~ AS e−iHt/~ .
Comparing these definitions with the ones given for the interaction picture, we see that
the difference between Heisenberg and interaction picture is that in the interaction picture
H0 instead of H is present.
Analyzing the time-evolution of |α, t0 ; tiI ,
i~
∂
|α, t0 ; tiI =
∂t
=
=
∂ iH0 t/~
e
|α, t0 ; tiS
∂t
−H0 eiH0 t/~ |α, t0 ; tiS + eiH0 t/~ (H0 + V ) |α, t0 ; tiS
e|iH0 t/~ V{ze−iH0 t/~} eiH0 t/~ |α, t0 ; tiS ,
{z
}
|
i~
VI
one has
i~
|α,t0 ;tiI
∂
|α, t0 ; tiI = VI |α, t0 ; tiI
∂t
(4.28)
And the time-evolution of an observable that does not contain time dependence explicitly
in the Schrödinger picture is
dAI
1
(4.29)
= [AI , H0 ] .
dt
i~
Reminder: Time-evolution
Heisenberg picture
Interaction picture
Schrödinger picture
quantum state
no change
evolution given by VI
evolution given by H
observable
evolution given by H
evolution given by H0
no change
4.5. TIME-DEPENDENT HAMILTONIAN: THE DIRAC PICTURE
Let us expand |α, t0 ; tiI as
|α, t0 ; tiI =
X
cn (t) |ni .
111
(4.30)
n
We substitute the sum (4.30) in eq. (4.28) and multiply by hn| on the left:
i~
X
∂
hn| VI |mi hm| α, t0 ; tiI .
hn |α, t0 ; tiI =
| {z }
∂t
m
(4.31)
identity
Using
hn| VI |mi = hn| eiH0 t/~ V (t)e−iH0 t/~ |mi
=
Vnm (t)ei(En −Em )t/~
and
cn (t) = hn |α, t0 ; tiI ,
we write eq. (4.31) as
i~ dtd cn (t) =
with
P
m
Vnm eiωnm t cm (t)
(4.32)
En − Em
= −ωmn .
~
Eq. (4.32) corresponds to a coupled set of differential equations:

 

ċ1
V11
V12 eiω12 t . . .
 ċ2   V21 eiω21 t

V22

 

i~  ċ  = 

..
3

 

.
V33
..
.
V44
ωnm =
c1
c2
c3
..
.
cn (t) gives the probability of finding state |ni as a function of t.



.

Consider the two-state problem with a sinusoidal oscillating potential by using the interaction picture.
→
H0 = E1 |1i h1| + E2 |2i h2|
V (t) = γeiωt |1i h2| + γe−iωt |2i h1|
(E2 > E1 )
(γ, ω ∈ R)
Probability of finding the system in one of the two states is
(
)
2
2 1/2
2
2
γ
(ω
−
ω
)
γ
/~
21
sin2
+
t
|c2 (t)|2 = 2 2
γ /~ + (ω − ω21 )2 /4
~2
4
|c1 (t)|2 = 1 − |c2 (t)|2
ω21 =
E2 − E1
~
← Rabi’s formula
112
CHAPTER 4. APPROXIMATION METHODS: PERTURBATION THEORY
Resonance
When the frequency of the external potential ω is equal to the characteristic frequency of
the two level system ω21 , we are in a resonance situation
ω ≃ ω21 =
E2 − E1
~
and
|c2 (t)|2 = sin2 γ~ t
|c1 (t)|2 = cos2 γ~ t
From t = 0 until t = π ~/2 γ, the system absorbs energy from the time-dependent potential
and makes a transition from E1 to E2 . In the interval of time between t = π ~/2 γ and
t = π ~/γ, the system makes a transition to the ground state (E1 ) and gives extra energy to
V (t). We have a cycle of absorption/emission processes. This phenomenon also happens
away from the resonance, but the probabilities decrease.
V̂ (t) works as a source and a sink of energy.
This example defines the mechanism of a MASER (microwave amplification by stimulated
emission of radiation).
Maser
The ammonia molecule NH3 has two parity states | Si and | Ai lying close together with
E A > ES .
4.5. TIME-DEPENDENT HAMILTONIAN: THE DIRAC PICTURE
113
S
and is in the miThe characteristic frequency for this system is given by ω21 = EA −E
~
crowave regime.
The molecule has an electric dipole moment defined by µ
~ el . In the presence of a timedependent electric field, the potential energy adquired by the molecule is:
~
V (t) = −~µel · E
with
~ = E0 ez cos ωt,
E
according to the correspondence principle, the perturbing hamilton operator can be defined
by V̂ (t) = µel,z E0 cosωt.
The condition of resonance is when ω = ω21 the frequency of the perturbing potential
coincides with the characteristic frequency of the molecule.
The idea of a MASER is to prepare a pure beam of ammonia molecules in the state |A >
S
. The dimension of the cavity is
which enters a microwave cavity tuned to ω = EA −E
~
114
CHAPTER 4. APPROXIMATION METHODS: PERTURBATION THEORY
such that the time spent by the molecules in the cavity is (π/2)~/γ. As a result, one is
in the first emission phase of the Figure ??, we have |A > in and |S > out.
The excess energy of | Ai is given up to the time-dependent potential as |A > turns to
|S >, and the microwave radiation field gains energy. In this way, we obtain microwave
amplification by stimulated emission of radiation.
4.6
Time-dependent perturbation theory
Apart from the two-level problem, usually there are no known solutions for c(t). For this
reason, we approximate c(t) by the perturbation expansion of the form
(2)
(1)
cn (t) = c(0)
n + cn + cn + . . . ,
(1)
(2)
where cn , cn , . . . are amplitudes of the first, second, ... order in the strength parameters
of the time-dependent potential. In eq. (4.32),
i~
X
d
cn (t) =
Vnm eωnm t cm (t),
dt
m
(0)
if at t0 = 0, only the state i is populated, we replace cm (t) by cm = δmi and relate it to
(1)
the time derivative of cn :
X
d
i~ c(1)
Vnm eiωnm δmi .
n =
dt
m
(2)
This procedure is iterated to obtain the differential equation for cn and so on.
4.6.1
Operator formalism: Dyson series
There is a more elegant way of performing time-dependent perturbation theory consisting
in the following:
- we consider the evolution operator in the interaction picture, UI (t, t0 );
- we define a perturbation expansion for UI (t, t0 );
- we relate the matrix elements of UI to cn (t).
115
4.6. TIME-DEPENDENT PERTURBATION THEORY
we define UI (t, t0 ) as
|α, t0 ; tiI = UI (t, t0 ) |α, t0 ; t0 iI .
Then, from the differential equation
i~
∂
|α, t0 ; tiI = VI |α, t0 ; tiI ,
∂t
i~
d
UI (t, t0 ) = VI (t)UI (t, t0 ) ,
dt
we have
(4.33)
with
UI (t, t0 )|t=t0 = 1.
(4.34)
Integrating eq. (4.33), we get the integral equation:
i
UI (t, t0 ) = 1 −
~
Z
t
VI (t′ ) UI (t′ , t0 ) dt′ ,
t0
which fulfills the boundary condition (4.34). If we iterate now this equation, we can obtain
an approximate solution for the evolution operator (remember the Schrödinger integral
equation in scattering theory!!):
#
"
Z
Z t′
i t
i
UI (t, t0 ) = 1 −
VI (t′ ) 1 −
VI (t′ ) UI (t′′ , t0 ) dt′′ dt′
~ t0
~ t0
and, proceeding in the same way,
i
UI (t, t0 ) = 1 −
~
+ ... +
Z
t
′
′
dt VI (t ) +
t0
−i
~
n Z
t
dt′
t0
Z
−i
~
t′
2 Z
t
dt
′
t0
dt′′ . . . ×
t0
Z
Z
t′
dt′′ VI (t′ )VI (t′′ )
t0
t(n−1)
dt(n) VI (t′ )VI (t′′ ) . . . VI (tn ),
t0
which is the Dyson series. Freeman J. Dyson was the one to introduce this expansion for
QED.
4.6.2
Concept of transition probability
If UI (t, t0 ) is known, the time development of any state can be obtained via
|i, t0 = 0; tiI = UI (t, 0) |ii =
X
n
|ni hn| UI (t, 0) |ii,
|
{z
}
cn (t)
where | hn| UI (t, 0) |ii |2 gives the probability of finding the initial state i in state n at time
t.
116
CHAPTER 4. APPROXIMATION METHODS: PERTURBATION THEORY
Properties of UI (t, t0 ). Since
|α, t0 ; tiI = eiH0 t/~ |α, t0 ; tiS
= eiH0 t/~ U (t, t0 ) |α, t0 ; t0 iS
= eiH0 t/~ U (t, t0 ) e−iH0 t0 /~ |α, t0 ; t0 iI ,
|
{z
}
UI (t,t0 )
we have
UI (t, t0 ) = eiH0 t/~ U (t, t0 ) e−iH0 t0 /~ .
(4.35)
From (4.35) follows that
hn| UI (t, t0 ) |ii = ei(En t−Ei t0 )/~ hn| U (t, t0 ) |ii
(|ni, |ii eigenstates of H0 ).
We see that though
hn| U (t, t0 ) |ii is the transition amplitude and
hn| UI (t, t0 ) |ii is not the transition amplitude,
| hn| UI (t, t0 ) |ii |2 = | hn| U (t, t0 ) |ii |2
is the transition probability.
4.6.3
Perturbation expansion
Now, we are going to exploit the relation between cn (t) and UI (t, t0 ),
cn (t) = hn| UI (t, t0 ) |ii ,
(0)
(4.36)
(1)
to find the expansion terms cn (t), cn (t), . . . of
(1)
(2)
cn (t) = c(0)
n (t) + cn (t) + cn (t) + . . .
(4.37)
comparing (4.37) with the Dyson series expansion for UI (t, t0 ). We get:
c(0)
n (t) = δni
Z
Z
−i t
−i t iωni t′
(1)
′
′
cn (t) =
Vni (t′ )dt′
hn| VI (t ) |ii dt =
e
~ t0
~ t0
2 X Z t Z t ′
−i
′′
′
(2)
dt′
cn (t) =
dt′ eiωnm t Vnm (t′ )eiωmi t Vmi (t′′ )
~
t0
t0
m
..
.
(4.38)
with ei(En −Ei )t/~ = eiωni t . Then, the transition probability for |ii → |ni with |ni 6= |ii is
given by
(2)
2
P (i → n) = |c(1)
n (t) + cn (t) + . . . | .
117
4.7. FERMI’S GOLDEN RULE
4.6.4
Example: A constant perturbation
V (t) =
0 for t < 0
V for t ≥ 0
V (t) = p~ · ~x, for instance.
We assume that at t0 = 0 the system is at |ii and, using (4.38), calculate the expansion
(0)
(1)
terms cn and cn :
Then,
= c(0)
c(0)
n
n (0) = δin ,
Z t
−i
Vni
′
(1)
cn =
eiωni t dt =
1 − eiωni t .
Vni
~
En − Ei
0
|Vni |2
(2 − 2 cos ωni t)
|En − Ei |2
4|Vni |2
2 (En − Ei )t
sin
.
=
|En − Ei |2
2~
2
|c(1)
=
n |
(1)
Plot of |cn (t)|2 for a given t, with ω ≡
4.7
En −Ei
~
Fermi’s golden rule
In a realistic case, a final state usually forms a continuous energy spectrum near Ei (see
Fig. 4.1).
Or, another example could be the deexcitation of an excited atomic state via emission of
an Auger electron. Consider helium in the configuration 2s2 , i.e., with both electrons in
118
CHAPTER 4. APPROXIMATION METHODS: PERTURBATION THEORY
Figure 4.1: Elastic scattering of a plane wave by some finite range potential
an excited state. The system can get deexcited via
He(2s)2 → He+ + free electron
The transition probability in this process is summed over final states with En ≃ Ei :
X
2
|c(1)
(4.39)
n | .
n,En ≃Ei
→ Definition: density of final state is the number of states within the energy interval
(E, E + dE): ρ(E)dE.
In terms of ρ(E), (4.39) becomes
Z
X
(1) 2
2
|cn | →
dEn ρ(En )|c(1)
n |
n,En ≃Ei
= 4
Z
sin
2
|Vni |2
(En − Ei )t
ρ(En )dEn
2~
|En − Ei |2
For t → ∞,
1
πt
2 (En − Ei )t
lim
δ(En − Ei )
sin
=
2
t→∞ |En − Ei |
2~
2~
and
lim
t→∞
Z
2
dEn ρ(En )|c(1)
n (t)|
=
2π
~
1 sin2 ax
lim
= δ(x)
a→∞ π ax2
|Vni |2 ρ(En )t|En ≃Ei ,
(4.40)
where |Vni |2 is the average of |Vni |2 over the final states |ni.
We see from (4.40) that for large t the total transition probability of having a final
state near Ei is linear in t. Consequently, the transition rate, defined as the transition
probability per unit time,
!
d X (1) 2
[n]: group of final states
|cn |
wi→[n] =
dt
n
119
4.7. FERMI’S GOLDEN RULE
is constant for large t:
2π
|Vni |2 ρ(En )En ≃Ei .
~
This is the
R Fermi’s golden rule. It can also be written after performing the integration
over [n], dEn ρ(En ), as
2π
wi→n =
|Vni |2 δ(En − Ei ) .
~
wi→[n] =
We also consider the 2nd order term:
2 X
Z t′
Z t
−i
′′
′ iωnm t′
(2)
Vnm Vmi
cn =
dt′′ eiωmi t
dt e
~
0
0
m
Z
t
i X Vnm Vmi
iωnm t′
iωni t′
=
−e
e
dt′ .
~ m Em − Ei 0
(1)
′
(4.41)
′
The eiωni t term in (4.41) has the same t-dependece as cn , and the eiωnm t term gives rise
to rapid oscillations when Em 6= En 6= Ei .
Now, putting the results for c(1) and c(2) together, we get
2
X Vnm Vmi 2π wi→[n] =
Vni +
ρ(En )|En ≃Ei .
~ Ei − Em (4.42)
m
Interpretation of the formula (4.42)
|ii makes an energy non-conserving transition to |mi;
|mi makes an energy non-conserving transition to |ni;
only between |ii and |ni there is energy conservation.
The energy non-conserving transitions are virtual transition:
→ the first-order term Vni is a direct energy-conserving real transition;
→ the second-order term includes transitions to virtual intermediate
states and is non-energy conserving.
If Em ≃ Ei , we add an imaginary term in (4.42),
Ei − Em → Ei − Em + iǫ,
as we did with the Lippmann-Schwinder equation.