Chapter 3	Chapter 4
• Operators • Eigenvalue equation • Momentum operator in position space • Hamiltonian operator • Delta function • Expectation value – [position space representation] • Gaussian wave packet • Time evolution, • & Time dependent Schrodinger eq. • Time evolution operator	• Particle in a box • Dirac Notation • Hilbert space – Inner product • Hermetian operators Problem 4.4
Class development: Start with Dirac notation and vector spaces Introduce formalism: operators, states, inner product Derive the position space representation & the wave function

	General Physical state where all possible information about the system is known.
	The general state can be a combination of any of the allowable states of the system.
	Physical state where specific observables [] are used to characterize the system state.
	For any observable there is a corresponding operator.
	Operators acting on physical states that possess a definite value for that observable satisfy and eigenvalue equation.
	Complex number reflecting the amount of any state that is included in . {The representation of a vector in terms of a sum of other vectors should clarify the meaning}
	Probability that the system will be found in the associated state if a measurement is performed.
ensemble	Probability means that repeated measurements performed on identical systems can be used to find the frequency and therefore the probability of an occurrence. Of course any givien experiment results in one definitive result.
	Generalized inner or dot product. {Vectors}
	Require the bra vector is an associated vector that has amplitudes that are complex conjugates of the original vectors. This is an extension of the inner product formalism crucial to the interpretation of QM.
	The ith vector of a set of vectors that span the space.
	Complete set of basis vectors for some quantum systems. [Quantum systems can get more complicated so that other variables such as spin may need to be added]. The observable labeling the state is the particles location.
	Basis using particle momentum as the observable that labels the states.
	Chosen vectors are orthonormal and discrete. is the kronecker delta
	Chosen vectors are orthonormal and continuous. is the Dirac delta
Measurement	An experiment designed to measure a characteristic of a system will return a distinct value as a result for any conceivable quantum system. However, the act of measuring will force the system into a quantum state that is characterized by this value. {Measure a system’s location and the result is xo then the physical system jumps into or becomes the quantum state .}

At this point there the time evolution of the system has not been incorporated into the formalism. We can ask about the present configuration but do not know what the system will look like at some time later. [Classically it is not surprising that knowing the current configuration doesn’t mean you know the future or past configuration. The eq. of motion will evolve the system. {Predict earth’s position in a few days.}

We introduce the notion that there is a quantum state that will have the property that a particle is at a specific location.

This would be an expected possibility based on classical physics and therefore should seem as a very plausible aspect of quantum mechanics. The rules for this state will need to be developed but the basic definition should be clear. A particle that is the quantum state is a particle located at x. Just as in classical mechanics one can ask-

If the particle is at x at some time will be there at some other time?

A relevant quantum question would be-

If the particle is at x do we know its energy or momentum?

The answer is that the postion and momentum of a particle cannot be simultaneously known.

To proceed with a quantum formalism we will introduce an operator. Called the position operator with the property that

At this point this is a completely ad hoc idea. Quantum mechanics need a different formalism in order to adequately describe small systems and the introduction of operators associated with observables is a fundamental but new aspect.

Next we introduce another new idea that a quantum state can be defined that is based on combining two quantum states.

This immediately reveals an interesting possibility. The above state seems to have the particle at two locations and .

The state is not a position eigenstate because

This new state is not the original state times a constant unless = 1. Notice the above state in brackets has the same coefficient for but a different coefficient for . The state is new because it has different admixtures of and than the original state. So we conclude that is not an eigenstate of position. So the particle is not in a definite location.

The interpretation is that the new state describes a situation where the particle has an amplitude to be at and an amplitude to be at . The probability that in such a system a measurement reveals that the particle is atis

and

Therefore we can build the state with a 50% chance of being measured at each location as

This provides some interpretation for the meaning of the state. We are allowing quantum particles to exist in states that are ill defined in terms of particle location. A particle can be in two places at once. However we are not concluding that we can directly measure this fact directly. We are stating that this will be reflected in a measurement by preparing many identical experiments and recording the measured particle location. After sufficient recording the results will show an equally likely chance that the particle is either at or . But any single measurement will record the singular result that the particle is at one location or the other.

Examine:

The inner product of a state with a normalized vector tells us the amplitude for that state contained in the general state.

This tells us that the inner product of a vector with itself is just the sum of the probabilities for all eventualities which must be 1. This is the convention that supports the interpretation or vice versa this is the requirement which allows the probability interpretation.

For the general state

Lets imagine that there are a set of discrete locations where the particle can be found so that the general state is a sum over the complete set of possible locations. The amplitude for each position state is labeled with and i. This would probably be considered as poor notation but it says that if we have a general state labeled by then we will label the amounts of this state for each of the possible basis states as . This choice is based on the notation used when the states are viewed as continuous.

	This is the 2-d analog. To illustrate the fact that we have two “associated” vectors I show the dot product in matrix form. This requires that you introduce a column and row vector representation of the vector. This is so trivial that it not mentioned when first introduced but we are extending the difference by requiring for the Hilbert space.

This is the normalization condition. The key point is that it is equivalent to the much simpler result obtained above for a simpler system.

We are going to extend this way of building states to the situation where we describe quantum states in terms of the amplitude for the particle to be at each location in space. Thus rather that providing two coefficients and we provide the function . This allows us to say that the particle is spread over all space with contributions from each location possibly different. To maintain a reasonable normalization we require

This should be recognized is the same approach taken above when assigning valuesto the coefficients. The integral is required because of the continuous nature of the locations x.

can be interpreted as a probability density.

let us examine the

What is the wave function [Position space representation].

Insert a complete set of states

For vector spaces one can always project a vector onto a compete basis.

For example:

Dirac notation makes this more transparent.

representation of an operator in position space. It would be a matrix if the states were discrete.