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4
Time evolution
In the previous sections we developed the formalism needed to describe a quantum spin and
the physical observables associated with such a system. In this section I want to finally
discuss the dynamics of a quantum spin system and determine how it evolves in time.
4.1
The Schrödinger Equation
We wrote the general state of the system in the ẑ-basis as,
| i = ↵ |"z i + |#z i
(4.1)
But, this gives the impression of a very static situation, surely the state of the system evolves
in time in some fashion. So, instead let’s write the general state with this time dependence
made explicit,
| (t) i = ↵(t) |"z i + (t) |#z i
(4.2)
On the LHS I put the time dependence together with as a way to remember that this
state depends on time. On the RHS I only made the coefficient time dependent, because the
eigenstates of the ẑ-spin operator, namely |"z i and |#z i do not depend on time. So really
what we are after is to understand how the coefficients ↵(t) and (t) depend on time.
There is nothing in what we have done so far to tell us how the state of the system would
evolve in time. It is one of the postulates of quantum mechanics that the evolution of
the state with time is determined by the Schrödinger equation. What is the Schrödinger
equation? You might have already seen it in a di↵erent context, but I want you to forget for
now what you have seen. The Schrödinger equation in general is an equation that relates
the time derivative of | (t) i to the action of the Hamiltonian on that state. So in general
the Schrödinger equation is given by,
Ĥ | (t) i = i~
@
| (t) i
@t
(4.3)
This might look very mysterious, but we will understand the meaning of this expression
p
in detail and through many examples. Let’s start with the RHS. The symbol i =
1 is
the imaginary unity, and its presence is essential to quantum mechanics. Unlike classical
mechanics, where complex numbers are sometimes employed for convience, the presence of
complex numbers is an essential part of the quantum dynamics. The symbol ~ is the reduced
Planck constant that we have already met. The derivative acting on the state should be
understand through its action on the coefficients ↵(t) and (t),
@
| (t) i = ↵(t)
˙
|"z i +
@t
40
˙ (t) |#z i
(4.4)
where ↵˙ = @↵/@t as usual represents di↵erentiation with respect to time. So again, the basis
states are independent of time, and it is only the coefficients that change with time and can
be di↵erentiated.
Now let’s consider the LHS of Eq. (4.3). I can again write it by expanding in the basis states
as,
✓
◆
✓
◆
Ĥ | (t) i = ↵(t) Ĥ |"z i + (t) Ĥ |#z i
(4.5)
Using the expansions of the LHS and RHS in terms of the basis states, I can now sandwich
both sides of the Schrödinger equation by h "z | to get,
✓
◆
✓
◆
↵(t) h "z | Ĥ |"z i + (t) h "z | Ĥ |#z i = i~ ↵(t)
˙
(4.6)
where I have used the orthonoramlity relations h"z |"z i = 1 and h"z |#z i = 0. I can similarly
sandwich instead with h #z | to get,
✓
◆
✓
◆
↵(t) h #z | Ĥ |"z i + (t) h #z | Ĥ |#z i = i~ ˙ (t)
(4.7)
These two equations look a bit daunting, but by simplifying the notation you would recognize
how simple they really are. We can write them as,
H"" ↵(t) + H"# (t) = i~ ↵(t)
˙
(4.8)
H#" ↵(t) + H## (t) = i~ ˙ (t)
(4.9)
where H"" , H"# , H#" , and H## are just complex numbers that in general also depend on time
and are defined as,
H"" = h "z | Ĥ |"z i
(4.10)
H"# = h "z | Ĥ |#z i
(4.11)
H## = h #z | Ĥ |#z i
(4.13)
H#" = h #z | Ĥ |"z i
(4.12)
⇤
(note that H#" = H"#
). Thus, we reduced the Schrödinger equation in this case to Eq. (4.8),
which is just two coupled ordinary di↵erential equations for ↵(t) and (t). Let’s build some
intuition by finding the solution in a few simple examples.
4.2
Constant magnetic field
Let’s start by placing the system in a constant magnetic field. I am always free to choose
~ = B ẑ. Then
my coordinate system so that the magnetic field is aligned along the ẑ-axis, B
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the Hamiltonian is given by,
~ =
~s · B
Ĥ =
Bŝz
(4.14)
Then, using the fact that ŝz |"z i = + ~2 |"z i and ŝz |#z i =
~
2
|#z i we can easily workout that,
~
B,
2
H"" = h "z | Ĥ |"z i =
H"# = h "z | Ĥ |#z i = 0 ,
(4.15)
(4.16)
H#" = h #z | Ĥ |"z i = 0 ,
(4.17)
H## = h #z | Ĥ |#z i = +
(4.18)
~
B.
2
The di↵erential equation (4.8) then simplifies to two uncoupled equations,
~
B ↵(t) = i~ ↵(t)
˙
2
(4.19)
~
B (t) = i~ ˙ (t)
2
The ~ drops out and the solution to these two equations is immediate,
+
↵(t) = ↵0 e+i!t/2
(t) = 0 e i!t/2
(4.20)
(4.21)
(4.22)
where ↵0 = ↵(0) and 0 = (0) are the initial values of the coefficients and ! = B is the
Larmor frequency we met in section 1 above, Eq. (1.40). Thus, the state as a function of
time behaves as,
| (t) i = ↵0 ei!t/2 |"z i +
0
e
i!t/2
|#z i
(4.23)
Interestingly, the probability of finding the state in the up-spin state or the down-spin state
is independent of time and is just set by the initial condition,
Pr("z ) = |↵(t)|2 = ↵⇤ (t)↵(t) = ↵0⇤ ↵0 = |↵0 |2
(4.24)
and similarly for (t). The phase of the spin-up and spin-down states changes with time
according to the Larmor frequency, but the probability of finding the system in one of these
states is independent of time.
4.3
Stationary states
The Hamiltonian is itself an operator and as such it has eigenstates and eigenvalues associated
with it. I will write these eigenstates and eigenvalues as,
Ĥ |
Ĥ |
1i
= E1 |
2 i = E2 |
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1i
2i
(4.25)
(4.26)
where E1 < E2 are the two energies of the system and the states are orthonormal, h i | j i =
ij . In a system with a larger number of dimensions there would be many more eigenstates
of the Hamiltonian and many more eigenvalues corresponding to the many di↵erent possible
energies of the system. Just like any other observable, we can use the eigenstates of the
Hamiltonian to write a general state as an expansion in terms of these eigenstates,
| (t) i = c1 (t) |
1i
+ c2 (t) |
2i
(4.27)
where just as before the coefficients are functions of time. In this basis the Schrödinger
equation is particularly simple. Sandwiching the Schrödinger equation on the left with h 1 |
I get,
c1 (t) h
1 | Ĥ
|
1i
+ c2 (t) h
Similarly sandwiching on the left with h
c1 (t) h
Finally, using the fact that |
have,
2 | Ĥ
1i
|
1i
and |
2|
|
2i
= i~ ċ1
(4.28)
2 | Ĥ
|
2i
= i~ ċ2
(4.29)
I find
+ c2 (t) h
2i
1 | Ĥ
are eigenstates of the Hamiltonian Eq. (4.25) we
E1 c1 (t) = i~ ċ1
E2 c2 (t) = i~ ċ2
(4.30)
(4.31)
The solution to this set of uncoupled di↵erential equations is immediate,
c1 (t) = c1 (0) e
c2 (t) = c2 (0) e
iE1 t/~
(4.32)
(4.33)
iE2 t/~
and the state of the system evolves according to,
| (t) i = c1 (0) e
i!1 t
|
1i
+ c2 (0) e
i!2 t
|
2i
(4.34)
where I defined the angular frequencies, !1 = E1 /~ and !2 = E2 /~. If the initial conditions
are such that c1 (0) = 1 and c2 (0) = 0 then the system just remains in state | 1 i. Similarly,
if the system starts in the second energy eigenstate with c1 (0) = 0 and c2 (0) = 1 then it
remains in that state. Only the phase of the state is changing in time, but that does not
a↵ect the probability. The energy eigenstates are therefore known as stationary states.
~ = B ẑ the eigenstates of
You will notice that in the case of a constant magnetic field B
spin, |"z i and |#z i, coincide with the energy eigenstates. This is not surprising since the
Hamiltonian Ĥ is proportional to the spin operator ŝz (see Eq. (4.14)). In that case the
energy eigenvalues we found in the last section were,
~
B
2
~
= + B
2
E1 =
(4.35)
E2
(4.36)
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and | 1 i = |"z i and | 2 i |#z i and so the energy eigenstates can actually be labeled according
to the spin orientation rather than the generic labels 1 and 2.
In the case of a two-dimensional Hilbert space, as in the example of a single-spin system, the
Hamiltonian is equivalent to a 2 ⇥ 2 matrix and thus it can always be diagonalized and its
eigenstates and eigenvalues (the energies) can always be found analytically. Although the
following example is rather trivial, let me work it out anyways so you see how it is done in
a simple case.
~ = B x̂ then the HamilSuppose we take the magnetic field to be along the x̂-direction, so B
tonian is,
Ĥ =
Bŝx
(4.37)
Let’s work out the matrix representation of the Hamiltonian in the ẑ-basis. Using the general
relationship we found between an operator and the corresponding matrix, Eq. (3.25), we have
✓
◆
h "z | Ĥ |"z i h "z | Ĥ |#z i
Ĥ
()
(4.38)
h #z | Ĥ |"z i h #z | Ĥ |#z i
Then, using the action of ŝx on the ẑ-spin states as we worked out in Eq. (3.8) we find that,
0 ~B ~B 1
@
()
Ĥ
2
2
~B
2
~B
2
The eigenvectors of this matrix are immediately clear
!
!
✓
◆
p1
p1
1
1
~B
2
2
= + 2~B
)
2
p1
p1
1 1
2
2
|
A
1i
(4.39)
()
p1
2
p1
2
!
(4.40)
()
p1
2
p1
2
!
(4.41)
and
~B
2
✓
1 1
1 1
◆
p1
2
p1
2
!
=
~B
2
p1
2
p1
2
!
)
|
2i
and the energy eigenvalues are as given by in the case of a constant magnetic field along
the ẑ-direction, Eq. (4.35). This is as it should be, the energy levels of the system should
not change just because we chose to describe the system with a di↵erent coordinate system.
Another way of seeing that this is in fact an equivalent description is to note that the energy
eigenstates corresponding to the eigenvectors we found are nothing but |"x i and |#x i,
|
|
1i
2i
=
=
p1
2
1
p
2
|"z i +
|"z i +
44
p1
2
1
p
2
|#z i = |"x i
|#z i = |#x i
(4.42)
(4.43)
4.4
Precession
In the case of classical magnetic moment we learned that the Larmor frequency is the rate
at which the dipole (or spin) precesses around a fixed magnetic field as in Fig. (1.5). This
picture is of course inadmissible in quantum mechanics because we cannot speak of the
all three components of the spin-vector simultaneously and so a mental image like that of
Fig. (1.5) is inappropriate. Nevertheless a remnant of this precession can be found in the
quantum motion as well as we shall now see.
~ = B ẑ along the ẑ-direction.
Let’s again consider a system with a constant magnetic field B
I can prepare the state of the system so that at t = 0 it is in the |"x i state, i.e. spin-up in
the x̂-direction. Then, in terms of the ẑ-basis the initial state can be written as,
| (0) i =
p1
2
|"z i +
p1
2
|#z i .
(4.44)
Since the ẑ-spin states are also the energy eigenstates, the evolution of the system in time
is straightforward, each spin state just evolves with a phase factor as in Eq. (4.23),
| (t) i =
p1 e+i!t/2
2
|"z i +
p1 e i!t/2
2
|#z i .
(4.45)
Here again ! = B is the Larmor frequency. Now, what is the probability of observing the
system in the spin-up state |"x i at time t?
The answer to this question is straightforward. Writing the state | (t) i in the x̂-basis,
| (t) i = ↵x (t) |"x i +
x (t) |#x i
(4.46)
the probability of observing the system in the spin-up state is according to the postulates
of quantum mechanics just |↵x (t)|2 . But, what is ↵x (t)? To calculate it we just have to
sandwich | (t) i on the left with h "x |,
↵x (t) = h"x | (t) i
⇣
p1 h "z | +
=
2
=
1 i!t/2
e
2
+ 12 e
p1 h #z |
2
i!t/2
⌘⇣
p1 ei!t/2
2
p1 e i!t/2
2
|"z i +
= cos(!t/2)
|#z i
⌘
(4.47)
where I have used the orthonormality of the basis states to obtain the last line. Thus we
have that,
Pr("x ) = |↵x (t)|2 = h"x | (t) i
= cos2 ( 12 !t)
45
2
(4.48)
We could similarly have asked what is the probability of finding the spin-down state if we
start the system in the spin-up state. That probability is given by,
x (t)
= = h#x | (t) i
⇣
p1 h "z | +
=
2
1 i!t/2
e
2
=
+ 12 e
p1 h #z |
2
i!t/2
⌘⇣
=
p1 ei!t/2
2
|"z i +
p1 e i!t/2
2
sin(!t/2)
|#z i
⌘
(4.49)
and so,
Pr(#x ) = sin2 ( 12 !t)
(4.50)
This is a curious result. It means that the system oscillates between being in the spinup state |"x i and spin-down state |#x i. This is reminiscent of a precession, and the rate of
oscillations is just the Larmor frequency as shown in Fig. 4.1. You can similarly work out the
probability of finding the system in the spin-up and spin-down states along the ŷ-direction,
|"y i and |#y i. You will again find a remnant of the classical precession with spin-up and
spin-down oscillating between each other with the Larmor frequency.
2.
probability
1.5
PrH≠x L = Cos2 HwtL
1.
PrHØx L = Sin2 HwtL
0.5
0
0
p
2
p
3p
2
2p
wt
Figure 4.1: A quantum system initialized in the spin-up state |"x i in the presence of a
~ = B ẑ would experience oscillations between the spin-up and spin-down
magnetic field B
states along the x̂ direction at the Larmor frequency ! = B.
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