Download 2.5 Spin polarization principle 2.6 The commutator

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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3 Particles
What we said above about spins can be generalized to any quantum mechanical system.
Let’s do this and then apply the principles to particles that have mass, position and
momentum
1. The quantum mechanical state of a system is defined by a complex vector |yi
which is an element of Hilbert space H. This space can have finite dimension of
infinite dimension. H is a generalization of C2 to higher dimensions.
2. The state vector |yi does not describe any physical properties of the system.
3. Instead, the state vector |yi describes the way a system behaves when a physical
property A with an experimental apparatus.
4. The state vector |yi determines the probability of measuring a possible value a of
the observable A.
5. When a measurement is performed, the system ends up in one of many potential
states |ni that all form an orthonormal basis of H. If the state of the system is |yi
before the measurement, the probability that it will end up in |ni is given by
|hn|yi|2 = hy|ni hn|yi = hy| Pn |yi
6. The physical, observable, properties of a system are described by operators A that
act on the state vector. Which operators these are is determined by the measurement apparatus.
7. Physical observables are defined hermitian operators. Their eigenspaces are orthogonal and their spectrum is real.
8. Any observable with eigenvectors |ni and eigenvalues ln , n = 1, 2, ... can we
spectrally decomposed
A =
=
 ln Pn
n
 ln |ni hn|
n
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9. The expectation (average) value of measuring A is given by
h Ai = hy| A|yi
= hy|
 ln |ni hn|
n
 ln |hn|yi|
=
!
|yi
2
n
if the system is initially in state |yi.
3.1 Infinitely dimensional Hilbert-spaces
All of the above is straightforward when we have a Hilbert space that is infinitely dimensional and we are dealing with a countable basis
|ni
n = 1, 2, 3, 4, .....
And when operators have a spectral decomposition
A =
=
 ln Pn
n
 ln |ni hn|
n
this means that we can “count” the different measurement outcomes, in other words
the observables that these operators correspond to are quantized. However, there are
obervables that generate measurement values that are continuous, for example the position or momentum of a free particle. When we deal with such situations and corresponding Hilbert spaces we must be able to also have bases that depend on a continuous index. so
|ni n = 1, 2, 3, 4, ... $ | x i x 2 R
When we have a state |yi and a discrete basis and hence
|yi =
=
 yn |ni
n
Â(hn|yi) |ni
n
=
 |ni hn|yi
n
where the probability amplitudes
yn = hn|yi .
The decomposition of a vector in a continuous basis becomes
ˆ
dx | x i h x |yi
|yi =
ˆ
=
dx (h x |yi) | x i
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In analogy to the above we can write
y( x ) = h x |yi
So this is a probability ampliute that depends on the continuous label x. We can also
see that in the discrete version we have a “Zerlegung der Eins”
1=
 |ni hn|
n
Likewise for a continous basis we have
1=
ˆ
dx | x i h x |
So all we really have to formally remember is replacing sums by integrals
ˆ
 $ dx
n
How about the scalar product of two vectors? in a discrete system we have
!
hy|fi = hy|
 |ni hn|
|fi
n
= Â yn? fn
n
In the continous case this becomes
✓ˆ
◆
dx | x i h x | |fi
hy|fi = hy|
ˆ
= dxy? ( x )f( x )
We can even think of the complex function y( x ) as the vectors themselves and the
above integral as a dot product between two different vectors. But remember that the
functions y( x ) depends on the choices of basis vectors | x i Hence they only represent
the state.
• The state of the system is defined by a vector |yi
• it is represented by the amplitude function y( x ) (in the provided basis)
This is much like a point in a plane x may have some representation ( x, y) in one coordinate system and ( x 0 , y0 ) in another coordinate system.
Additng to the above, in a discrete system the normalization condition on a state
vector is given by
 | h n | y i |2
n
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