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Quantum
New way of looking at our world
Classical vs Quantum
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Typically a student develops an intuition about how the
world works using classical mechanics and practical
experience.
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Newton’s laws
Objects dropping
There are some necessary realignments but students
tend to accept much of what is explained and integrate it
into there world model
Quantum mechanics requires a completely new
formulation.
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The connections with practical experience are more difficult but
not impossible.
There is a tendency to be much more demanding at this stage.
Students are less likely to believe the principles and ideas.
QM Require a CHANGE in
Perception
State of the system
| STATE >
Labeled by the observables
| mass, momentum, energy >
Allows us to predict the results of measurements
STATE
Classical system
| mass,x,y,z,vx,vy,vz, Q >
Given a particle in a potential well there is an infinite set of
possible orbits. The mass and charge of course pick
certain orbits by choosing the strength of the potential
and of course the inertial response of the particle. The
position and velocity specify which of these classical
orbits the particle will follow. Thus given the value of
these variables and of course the forces (potential), the
orbit is determined and the location and velocity at any
time can be found.
Our states will be elements of a vector space. That means several important
things. Most of the properties of vector spaces can be understood by
considering the well know space of 3-vectors.
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There is a mathematical process called the dot product.
Vectors have components that determine (STRENGTH)
the amount in any direction.
Vectors can be expressed in terms of a basis.
Basis vectors need to be complete. You need to pick
basis vectors that allow you to specify all directions.
Usually this is x,y,z.
There are infinite possibilities for the basis vectors.
You can create mathematical relationships that define
how the components change. (Transformations &
Representations).
Quantum variation
For every observable we define an operator.
Aˆ | a   a | a 
The state is specified by a value of the observable
a.
**********Eigenvector-Eigenvalue**************
True for only a subset of all the states.
More on operators
The most general way that an operator can
act on a state is to transform it to a new
state.
Aˆ |     |  
Not all operators represent observables of
the system.
Rotation is an OPERATION
Pˆ  momentum
xˆ  position
Hˆ  energy
Lˆ  angular momentum
Sˆ  spin
Jˆ  Total angular momentum
iHˆ t
ˆ
U  time evolution  e
iJˆ
ˆ
R  rotation  e
|   i | i 
Dot Product
| 
Basis vectors
|   i | i 
|i 
|   i | i 
|   i | i 
For now we note
[ xˆ, Pˆ ]  i h
is the commutator for position and momentum. This equation determines the
relationship between the observables and makes position and momentum
mutually exclusive in the sense that both cannot be simultaneously precisely
measured for a quantum system.
Abner Shimony
1 State
“Associated with every physical system is a
complex linear vector space V, such that
each vector of unit length represents a
state of the system. “
Abner Shimony
2 Observables
“There is a one to one correspondence between
the set of eventualities (observables) concerning
the system and the set of subspaces of the
vector space associate with the system, such
that if e is an eventuality (observable) and E Is
the subspace that corresponds to it, then e is
true in a state |S> if and only if any vector that
represents S belong to E; and is false in the
state S if and only if any vector that represents S
belongs to E(orthogonal) .” A states described
by a vector that has components in both
subspaces represents a state with an
unspecified value for this observable e.
Abner Shimony
3- Measurement
“If |S> is a state and e is an eventuality
(observable) corresponding to the subspace E,
then the probability that e will turn out to be true
if the initially the system is in state |S> and an
operation is performed to actualize (measure) it.
probS (e)  || PE v || 2
v is a unit vector representing a state with an
observable value e … and PE is the projection.”
Abner Shimony
4 composites i.e. atom e,P
“If 1 and 2 are two physical systems, with
which the vector spaces V1 and V2 are
associated, then the composite system 1 +
2 consisting of 1 and 2 is associated with
the tensor product V1 x V2.”
Abner Shimony
5 Time evolution
“If a system is in a nonreactive environment
between 0 -> t, then there is a linear
operator U(t) such that U(t) |v> represents
the state of the system at time t if |v>
represents the state of the system at time
0. Furthermore, ||U(t) v||2 = || v||2 for all v
in the vector space.”