Download Lecture 7: More general Hilbert spaces

• in QM an observable A is represented by a hermintian operators A. When we
measure the observable the system will end up in one of the eigenstates of A and
the value that we measure is the corresponding eigenvalue.
• So say we have A = A† in C2 with eigenvectors |l1 i and |l2 i and corresponding
eigenvalues l1 and l2 (both real) then when we measure the observable A of a
system in state |si the instrument
– spits out a value l1 with probability p(l1 , A) = hs| Pl1 |si
– spits out a value l1 with probability p(l2 , A) = hs| Pl2 |si
Now remember the spectral decomposition of any hermitian operator
A = l1 Pl1 + l2 Pl2
Let’s apply this to the measurements of spin. It implies that for example we can consider the “observable” sz as a hermitian operator with eigenvectors |ui and |di and
eigenvalues h̄/2 and h̄/2 so
sz |ui =
sz |di =
h̄
|ui
2
h̄
|di
2
But that means that we can write
sz =
=
=
h̄
(|ui hu| |di hd|)
2
h̄
( Pu Pd )
2✓
◆
h̄
1 0
0
1
2
Likewise
h̄
h̄
sx = (+ ) |r i hr | + (
) |l i hl |
2
2
h̄
=
( Pr Pl )
2✓
◆
h̄
0 1
=
1 0
2
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and
h̄
2
h̄
2
h̄
2
h̄
2
h̄
2
sy =
=
=
=
=
The matrices
✓
0 1
1 0
◆
✓
,
( Pi
Po )
( Pi
1 + Pi )
(2Pi
✓
✓
0
i
1)
1
i
1
0
i
0
0
i
i
i
◆
1
◆
◆
✓
1
0
0
1
◆
are called Pauli spin matrices. When multiplied by h̄/2 they represent the observables
spin in x and y and z direction. Each Stern gerlach apparatus oriented in the unit directions measures these observables.
2.4.4 General apparatus orientation
Originally we worked with Stern Gerlach apparatus oriented in orthogonal directions
which have us three ways to measure spins, i.e. measurables sx , sy and sz . Now what
happens if we orient the apparatus in any direction
n̂ = (n x , ny , nz )
where this is the unit vector that is pointing in a direction given by those components.
It turns out that for this direction we have an observable that is given by
sn = n x sx + ny sy + nz sz = s · n̂
where we have a vector of operators
s = (sx , sy , sz )
Explicitely we have
sn =
=
✓
◆ ✓
◆ ✓
h̄
0 nx
0
ny i
nz
+
+
nx 0
iny
0
0
2
✓
◆
h̄
nz
n x iny
n x + iny
nz
2
0
nz
◆
Now let’s assume we prepare the system in state |ui by measuring sz and finding l =
h̄/2. Let’s then tilt the apparatus by an angle q towards the y axis so
n x = sin q,
ny = 0,
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nz = cos q
so in this case the apparatus is designed to measure
✓
◆
h̄
cos q
sin q
sn =
sin q
cos q
2
What we have said is that now performing a measurment will prepare the system in an
eigenstate of sn . So the eigenvalues of sn are given by
✓
◆
✓
◆
cos q/2
sin q/2
| v1 i =
| v2 i =
sin q/2
cos q/2
with eigenvalues
and
l1 = h̄/2
h̄/2
l2 =
what are the probabilities of measureing ± h̄/2? We can write
|v1 i = cos q/2 |ui + sin q/2 |di
sin q/2 |ui + cos q/2 |di
| v2 i =
p(sn , h̄/2) = hu| Pv1 |ui
= hu| (cos q/2 |ui + sin q/2 |di) (cos q/2 hu| + sin q/2 hd|) |ui
cos q/2 ⇥ cos q/2
= cos2 q/2
and
p(sn ,
h̄/2) = sin2 q/2
So when q = p/4 we have
p(sn , h̄/2) = 85%
p(sn ,
h̄/2) = 15%
2.4.5 Expectation values
The above example illustrates that it may also make sense to compute the average of a
spin. we would have
h̄
h̄
h̄
⇥ 0.15 = 0.7 .
hsn i = ⇥ 0.85
2
2
2
In general, let’s say we have prepared a system in state |si and perform a measurement
on observable A all the numerical values for these measurements are eigenvalues of A,
let’s call them l1 and l2 so the average is
h A i = l1 p1 + l2 p2
where p1 and p2 are the probabilities of measuring l1 or l2 respectively (and preparing
the corresponding eigenstates |v1 i or |v2 i). Remember that these probabilities are given
by
p1 = hs| P1 |si = hs|v1 i hv1 |si
p2 = hs| P2 |si = hs|v2 i hv2 |si
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so that
h A i = l1 p1 + l2 p2
= l1 h s | v1 i h v1 | s i + l2 h s | v2 i h v2 | s i
= hs| (l1 |v1 i hv1 | + l2 |v2 i hv2 |) |si
Now remember that the sanwiched stuff is the spectral decomposition of the operator
A so that
h Ai = hs| A|si
that means the expectation value of the measurement of A is the corresonding operator
in the middle of the bracket of the the incoming state. For example, let’s check this with
|si = |ui the incoming state prepared by measuring sz and selecting the outcome that
gives h̄/2. Now let’s measure
✓
◆
h̄
cos q
sin q
sn =
sin q
cos q
2
as above. On average we will measure a spin
hsn i = hu|sn |ui
✓
h̄
cos q
= (1, 0)
sin q
2
h̄
=
cos q
2
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sin q
cos q
◆✓
1
0
◆
2.5 Spin polarization principle
An interesting feature about the physics of spins is the following statement. Any normalized vector | ai in C2 is an eigenstate of some spin operator
sn = s · n̂.
In other words, for an arbitrary spin state of a system we can find an orientation n̂ of
the measurement apparatus that will give an experimental outcome of h̄/2 with 100%
certainty. Recall that
✓
◆ ✓
◆ ✓
◆
h̄
0 nx
0
ny i
nz
0
sn =
+
+
nx 0
iny
0
0
nz
2
✓
◆
h̄
nz
n x iny
=
n x + iny
nz
2
Now, sn is Hermitian, so it must have an orthonormal basis. So clearly if | ai is one of
those eigenvectors then with 100% certainty a measurement of sn will give the corresponding eigenvalue. One can show that the eigenvalues of sn are ± h̄/2.
2.6 The commutator
When we started talking about operators that act on vectors in vector spaces we saw
that the order at which operators are applied mattered and that generally we have
ABv 6= BAv
where A,B are two linear operators and v is a vector in some vector space. We discussed
this using the example of a rotation by 90 in R2 and a projection onto the x-axis. We
might ask, what is the condition for hermitian operators in C2 for this to be actually
equal. Well let’s assume that both operators have identical eigenspaces
A=
 an |ni hn|
and
n
B=
 bn | n i h n |
n
where an and bn are eigenvalues of A and B respectively. In this case
AB =
=
 a n | n i h n |  bm | m i h m |
n
m
  a n bm | n i h n | m i h m |
n
=
m
 a n bn | n i h n |
n
= BA
This means if two operators have the same eigenvalues the order at which they are
applied is the same. Another useful way of saying this is
[ A, B] = 0
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where
[ A, B] = AB
BA
is known as the commutator. Physically this means that the two observables that
correspond to µA and B can be measured simulateneously. We say with the Stern
Gerlach experiments that this isn’t the case for the different components of spin. In
fact all operators si had different eigenspaces. When we look at the spin operators
Sx = h̄sx /2,Sy = h̄sy /2,Sz = h̄sz /2, with the Pauli matrices
✓
◆
✓
◆
✓
◆
0 1
0
i
1 0
sx =
, sy =
sz =
1 0
i 0
0
1
We can compute the commutator
✓✓
◆✓
h̄2
h̄2
0 1
0
[ S x , Sy ] =
[sx , sy ] =
1 0
i
4
4
◆ ✓
◆◆
2 ✓✓
h̄
i 0
i 0
=
0
i
0 i
4
✓
◆
h̄2
i 0
=
0
i
2
= ih̄Sz
0
i
◆
✓
0
i
0
i
◆✓
0 1
1 0
◆◆
Likewise we can show that
[Sy , Sz ] = ih̄Sx
[Sz , Sx ] = ih̄Sy
so indeed the components of spin cannot be measured simultaneously.
2.7 Spinor
Here’s another important viewpoint of a quantum mechanical spin. Remember that
any spin state can be written as a superposition of any basis
| s i = a1 | v1 i + a2 | v2 i
where e.g. the basis vectors |vi i could be the pair |ui and |di or the pair |l i and |r i or
any orthonormal set of basis vectors. When written this way we have
| s i = a1 | v1 i + a2 | v2 i
= (hv1 |si) |v1 i + (hv2 |si) |v2 i
2
=
 | vi i h vi | s i
i =1
and the quantities
h vi | s i
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were called probability amplitudes, because the probability of preparing a system in
state |vi i when an observable is measured for which this vector is an eigenstate is given
by
Pi = | hs|vi i |2
Let’s think of the vectors |vi i as fixed then the state of the spin is specified by the amplitudes
( a1 , a2 )
so a pair of complex numbers that fullfill the condition
 ai? ai = 1
i
If we think of the basis vectors as fixed and set in stone then every quantum spin,
is specified by the above pair, which is called a spinor. Note though that the actual
numbers are determined by the choice of basis vectors this is wky we say that
• The state of the quantum spin is DEFINED bz the vector |si
• but it is represented by the spinor (a1 , a2 ) in the basis |vi i
2.8 Time passes
Everything we have said so far is about states that are static, apart from what happens
during a measurement. We will learn soon that generally states evolve over time. For a
spin state this means that
|si = |s(t)i
what this means is that if we express the state as a superposition of basis vectors that
we consider static, the time dependence must be in the coefficients.
|s(t)i = a1 (t) |v1 i + a2 (t) |v2 i
Later on, we will see what physical laws govern the time evolution of these coefficients
and thus the time evolution of a quantum mechanical state. In fact we will derive an
ODE for the time evolution that looks like this
ih̄∂t |s(t)i = H |s(t)i
where H is an operator that corresponds to the obervable energy of a system. When
we look at this equation we see that the change in time is equivalent to applying the
operator H to system
ih̄∂t $ H
This is a very fundamental connection between energy and time. A similar one we will
learn about later, the connection between space and momentum
ih̄∂ x $ p x
ih̄∂y $ py
ih̄∂z $ pz
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or together
ih̄r $ P
Let’s say the above equation is true, what does this imply for the mean value of measurements of a general spin observable S? Remember that
hSi (t) = hs(t)|S|s(t)i
What’s the ODE for that quantity?
∂t hSi (t) = (∂t hs(t)|) S |s(t)i + hs(t)| S∂t |s(t)i
H
H†
= hs(t)| i S |s(t)i + hs(t)| Si |s(t)i
h̄
h̄
i
=
hs(t)| ( HS SH ) |s(t)i
h̄
i
∂t hSi =
h[ H, S]i
h̄
This is also a very important relation which we will come back to.
2.9 Summary
So, we have learned the following
1. The quantum mechanical state of a spin is defined by a two dimensional complex
vector |si which is an element of C2 .
2. The state vector |si does not describe any physical properties of the system.
3. Instead, the state vector |si describes the way a system behaves when a physical
property such as sx , sy , sz of generally sn is measured with an experimental
apparatus.
4. The state vector |si determines the probability of measuring a value of the observable.
5. When a measurement is performed, the system ends up in one of two states
| ai , |bi which are orthonormal vectors in C2 . If the state of the system is |si before
the measurement, the probability that it will end up in | ai is given by
|hs| ai|2 = hs| ai h a|si = hs| Pa |si
6. The physical, observable, properties of a spin are described by operators that act
on the state vector. Which operators these are is determined by the measurement
apparatus, for instance the orientation of the Stern Gerlach setup. Generally such
an observable has the form
s · n̂
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7. Physical observables are hermitian operators. Their eigenspaces are orthogonal
and their spectrum is real.
8. Any spin observable with eigenvectors | ai and |bi and eigenvalues l a and lb can
be written as
A = l a Pa + lb Pb
9. The expectation (average) value of measuring A is given by
h Ai = hs| A|si
if the system is initially in state |si.
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