Exercises in application:

Notation using a standard n-dimensional space.

Vector:

Inner product:

Orthonormal vectors:

Basis:

Vectors general:

Now let us define some linear combination of basis vectors:

This is equivalent to defining a state

Let us define n of these

Now define them such that these states are orhonormal

The basic manipulation of QM requires

• the sum of states be a state
• projection (inner product)
• orthogonality

Quantum states behave like a vector space with an inner product.

Interpretation:

The projection of one state onto another is an amplitude and just as in standard vector spaces provides an amount. Given a general state how much of this state involves the particle being at location x. An amplitude is complex and must be squared to interpret it as a probability. The complex nature of amplitudes allows for constructive and destructive interference. Operators can be defined that change vectors into vectors. If we require that the new vectors are simply linear combinations of the old vectors then the operators can be represented as matrices or written as

Certain operators in QM will be identified with measurements and are referred to as observables.

• Eigenstates are in a definite observable state
• Several measurements can be simultaneously made if the observables are compatible.
• A state will be labeled by a complete set of observables
• Some measurements are incompatible and a measurement of one precludes a measurement of the other without disrupting the first measurement.
• The eigenstates for an observable form a complete set (span the space). These states can be normalized and a suitable orhtonormal set can be found.
• Thus , , are bases for some physical situations.
• Integrals are for continuous eigenvalues. Sums are for discrete eigenvalues. Discrete point to a finite region (Standing waves on a string are the Fourier series while standing waves on an infinite string are treated with the Fourier transform.)

Consider a particle at location x1=50 and x2=200.

For an idealized case we can represent this state as:

andtells us the amount of each. Let us assume equal likelihood for each position.

Step 1 Normalize the state.

Must be very careful as to the definitions of , and

x1 and x2 are particular values while x is a continuous parameter.

where is an arbitrary phase angle. As a matter of fact we could have chosen the phases of both states to be different. A general solution is

The probability of finding the particle at x1 is

The expectation value for x

Probability of measuring a particle with momentum p1?

This result shows interference between the contribution from x1 and x2 to the state of definite momentum p1. Let .

To further interpret this result let us consider simply the state x1.To illuminate the important aspects we choose the normalization to be simply 1.

The result is finite. The result is independent of the choice of p1. This means that each momentum should have the same chance of being measured as part of the localized state, as we encountered in the appendix.

Conclusion:

• Momentum and position eigenstates cannot be properly normalized in the traditional sense. These states will lead to Dirac delta functions and sine and cosine functions that cover infinite ranges and therefore are completely spread out. Particles in pure momentum states have an exact momentum (delta function) are equally likely to be everywhere (sin(kx)). Particle in exact locations are spread over momentum space.
• Normalization of these states is determined by unitarity. The states normalized properly can be used as basis vectors that span the space.

Restated the normalization of the momentum states preserves the unitarity of the momentum and the position operators but cannot provide a finite interpretation because the delta function is composed of an number of momentum states.

On further note for these states: This bit is very confusing based on the abstract nature of the continuum. At this point one might ask why do we insist upon continuity? Perhaps space is discrete. However describing motion as a series of hops or understanding observables such as energy and momentum in this framework is also fraught with complexity. Lattice calculations that try to discretize space in order to calculate encounter a number of similar conundrums. Let us proceed to find a represenstation for the momentum operator wrt the position basis.

If forms a complete basis then any linear operator is determined if the mapping or amount of the state that becomes p(x,x’) is given. This is identical to writing a rotation in terms of a matrix. The elements of a matrix tell us how much j there is in i.

For the state

Remember:

• is an idealized state of a particle at an exact location
• is an idealized state of a particle with an exact momentum
• is a combination of all momentum states with equal weight for each state
• is a combination of all position states with equal weight for each state
• Thus there can be no weight ascribed to these contributions since it tends to zero.
• The wave functions for these states are delta functions ,
• The normalization used for the momentum and position state representations for the above states is finite and is determined by unitarity.

These states represent an idealized physical state. Here we are asking about a particle that is localized to infinite precision to a point. From any standpoint the possibility of a point raises difficulties. It should not be surprising that quantum mechanics can approach the reality as some type of limit. Indeed the delta function is the limit of a distribution and as long as we use states with some finite space extension the results are easier to understand. The standard example is to consider:

is the particle location; is some measure of the extension.

A Gaussian function does extend to infinity but the tails are small and the wavefunction can be used to treat a particle localized around with a spread .

Is this state a position eigenstate ? NO

Is it a momentum eigenstate? NO

Can we find the average value of x

Now what are the momentum properties of the state?

Find the expectation value for momentum

In the momentum basis:

In the postion basis: