Reminder:

Waves classically are a disturbance of a medium. Drop a pebble into a pond and the resulting circle wave will propagate as circular ripples away from the point of creation. These ripples can simultaneously arrive at two slits or apertures.  The disturbance arrives at the slits but the underlying medium is different. The water that rises and falls at one aperture is different than the water that rises and falls at the other.  Thus wave phenomena exhibit interference, spreading, propagation but do not violate intuition.  Quantum systems exhibit these wave characteristics as a fundamental behavior.  The statement that a fundamental entity somehow is characterized by being spread out is a new feature.  Classical wave properties need to be imported into the quantum formalism. On the other hand the discrete nature of particles is still an important property.  While we somehow need to imagine an electron passing through two different slits simultaneously we need to preserve the discrete “bullet-like” nature of a particle.  Our interference pattern on the screen arises from a sampling of discrete impacts. The electron hit here and the here and so on. Each measurement consists of a single whole electron striking a screen at one distinct location.

 

Chapter 3 introduces the formalism or framework that we will be developing to describe quantum systems.  In order to be able to describe mathematically the results we discussed for the two slit system we must have a new approach. Here are a few important elements.

  • State: A quantum system can be considered to be in a certain state. This state must reflect properties of the system and contain the information that is available about the system.
  • Observable: A measurable quantity. A particular quantum system can be found to be in a certain location x, have an energy E, have a momentum p or an angular momentum L although not all observables can be simultaneously measured.
  • Operator: An operator acts on an entity to produce another entity.  In quantum mechanics operators operate on quantum states.  Operators will be identified with observables.  The ability to measure a quantity implies the existence of an operator corresponding to that quantity.  The mathematical formalism will provide quantum systems with the features that explain how a system can have more than one value for an observable and provide the platform for developing the quantum or discrete nature of a measurement.
  • Vector spaces: Quantum states behave according to the same rules as vectors.
    • QS can be added and the result is a new quantum state.
    • Any sum that produces the same QS is equivalent in terms of its quantum properties.
  • Measurements: A measurement of an observable results in a particular value for that observable and forces the system to transition to a new quantum state that will posses the ability to have a distinct value for the measured observable. A state known to be spread out will become a localized state if a apparatus records the location of the system.
  • Amplitudes: States need to have amplitudes that characterize their amount. The use of an amplitude as distinct from the measured outcome will provide a mechanism to support interference.
  • Certain measurements cannot be performed simultaneously.  For waves this is not surprising. It is impossible to produce a sound that is an impulse in time and require that the impulse be characterized by a frequency.  Frequency is the regular repetition of a pattern over a time span.  This requirement of repetition over time precludes the possibility that the sound is instantaneous.  The requirement that the relationship between position and momentum be similar, i.e. inconceivable for a system to posses both definite characteristics, is new and surprising. 

 

 

Let  be a quantum state where a particle is in a definite location.  We mean a measurement of this state will find a particle at x.  To formalize this let us introduce an operator that will allow us to predict what a measurement will reveal for a given state.

 is a quantum state where a particle is at x.

 is an operator that mathematically operates on a quantum state.

To provide another example, let  be a quantum state where a particle has a definite momentum.

 is a quantum state where a particle has momentum p.

 is an operator that mathematically operates on a quantum state.

T

We will introduce a reasonable rule for the operation:

 

; operating on the state    just returns the state times x.

; operating on the state    just returns the state times p.

 

 

This is called an eigenvalue equation. The form is special and it doesn’t represent the most general action of an operator. The characteristic of an eigenvalue equation is that the original state under the action of an operator is replaced by the same state times a constant.

 

To clarify the notion of an operator we will consider the following two examples:

  1. rotation
  2. differentiation

 

1111111111111111111111111111111111

 

 

The matrix which represents the rotation operator changes the vector (x,y) to (x’,y’).  This is not in general an eigenequation because the new vector is not a simple multiple of the original but a completely new vector with new a new direction. However 3-d rotation about an axis leaves vectors along the axis unchanged. So some rotations leave some vectors unchanged.

 

2222222222222222222222222222

 

the partial derivative operator changes the function f to the function g. Again in general the new function g(x) is not a multiple of the original. However consider the following:

 

There are functions that become multiples of the original function after the application of the derivative operator.

 

 

An operator takes an entity of some type [vector, function, quantum state] and produces a an entity of the same type. 

vector è vector,

functionè function,

quantum state è quantum state.

 

So the general equation for an operator:

 ; state  becomes  times a constant under the action of .

 

Another property that will be required in the formalism is that quantum states can be added. The resulting state will be a quantum state.

 is a quantum state where a particle is at x1.

 is a quantum state where a particle is at x2.

 

 

One might guess that this state describes a state with a particle both at location x1 and at location x2.  The constants  somehow telling us how much of the stateis made up from the particle at location x1 and how much due to the particle at x2.

 

This notion can be extended to the continuum where a quantum state is described by providing the amplitude for the particle to be at any location x.

 

A quantum state  can be considered as a linear combination of states where the particle is known to be at a specific location for each of these states. When the sum is over a set of discrete states we use standard addition but the extension to the continuum will require the sum become an integral.  is the contribution or amplitude for the particle to be at a location x.

 

The above mathematics is reflected in the rules for vector addition.  Cleanly recognizing the vector nature of quantum systems will be essential for a sound understanding.  Crystallizing the relationship between well known three vectors and the Hilbert vector space of QM will provide keen insight.

 

This state may evolve in time and we find that:

 

Ψ(x,t) è wave function that contains all of the available information to describe the state of a system. 

; where Ψ(x,t) represents the amplitude for the particle to be at a location x at a given time t.

 

Just as a completely localized snap which represents a disturbance at a distinct location can evolve into a disturbance that spreads in a spherical pattern and that spherical pattern maintains some of the localized character and at the same time spreads out, so a quantum state can evolve in time with the result that the states characteristics in terms of a potential measurement will change or evolve.

 

 

In classical physics the equations of motion can be solved to produce a set of orbits or trajectories.  The specification of the system is done by providing initial values.  In quantum mechanics the state of the system contains all the knowledge available consistent with known observables but not all observables can be simultaneously measured.

 

 

CLASSICALLY

QM

1

Particle properties: mass, charge

Same

2

Particle observables: position, velocity

Same

3

Any observable available for any state

Only a subset of observables available

4

Particle interaction: Force or Hamiltonian

Same

5

Trajectories: Solutions to eq. of motion

Formalism provides different solutions

6

Conditions: Position and velocity at to

A subset of QM consistent observables

7

State: Trajectory consistent with conditions

Same but the trajectories are different (see 5)

 

 

To further the model we will treat the amplitude according to the same rules we discussed for wave:

  • Amplitude can be complex; ψ can be a complex function.
  • Probability is associated with the magnitude squared: ψ* ψ.

 

Ψ* Ψ è probability density that the particle is between x è x +dx

 

Observables:

Quantum and classical systems have observables or things that can be measured. Typically these are the same and there is a correspondence principle for how the quantum observable transitions to large scale classical physics.  A few phenomena do not have a classical analog. Fermion fields for example because of the exclusion principle do not have simple classical analogs. Particle spin can also be considered to be a purely quantum effect.

 

 For every observable there will be an operator.

 

Observable

 

Position

x

Momentum

-iћ∂x

Energy

-ћ2∂2x/2m  +V

Angular momentum

 

Velocity

 

Spin

 

 

 

 

The book will develop the representation of quantum systems in terms of the amplitude or wavefunction that describes the amount of state at any location.  In the following this particular choice results in the following rules for amplitudes.

 

 

A measurement that records the exact value of an observable forces the quantum state to become an eigenstate of this observable.

 

What is known about any observable in a particular quantum state is ascertained by expectation values. Two important expectation values are:

 

 

Of course eigenstates have a specific value for the corresponding observable and therefore σ=0 for that observable.

 

It is important to realize that a general quantum state accessible to a particle under the influence of a force is very broad.  Our focus will often be on the eigenstates but remember that these are special states.

 

In the study of sound waves it is convenient to talk about sinusoidal oscillation of a definite wavelength and frequency.  However a general sound wave has an almost arbitrary pressure distribution is space and time.

 

In addition to observable there are a few characteristics that are not allowed to change in our non-relativistic quantum theory. At this point we can label them as properties. A particle’s mass or charge is an example of a characteristic that we will consider as a property. No operators will be introduced to describe particle properties at this point.

 

TRANSLATION

 

One can imagine the transformation of a function such that f(x), g(x), h(x) are all the same function but at different locations. f(x) can be translated so as to obtain g(x) and h(x).

 

 

 

Using the Taylor series we can see that

 

 

using the expansion of ex

 

 

the expression

 

 

also

 

The state f(x) is moved as shown by the state labeled h(x) in the above figure.

 

 

 

To be clear we can consider two similar operations. The first shown above we refer to as translation. It is the operation of defining a new function h(x) that is the old function f(x) moved to a new position. h(x)=f(x-c).  In order to define the new function h(x) we can evaluate the old function f at the location x-c (previous location). So we need to find f(x-c) starting at x and moving by an amount c backwards.  h(x) is the translated f(x). 

 

The second operation we will call evolution. In this case we are looking to evaluate a function at some time c later knowing the function at time to.

 

f((t- to)+ to )=f(c+ to)

 

Since the Taylor series simply evaluates the function at different value given the change in the value, it is valid for the evolution of the state or for the translation of the state.  Translation is evaluating before and evolution is evaluating later. The sign in the exponent will change.

 

Where here the evolution is in x so we are finding the value of the function at x+ε knowing the value at x.

 

By defining the time derivative to be equivalent to the energy, as is done in the Schrödinger equation, we are specifying how the state evolves in time.

 

 

 

Because the energy operator is equivalent to the time derivative the time evolution operation for these states and these states only becomes.

 

 

All quantum states must satisfy the Schrödinger equation so time evolution and energy are related.