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Probability Interpretation

To make the language precise, we need to attach numerical values to the complex numbers introduced in rule II. To determine a complex number we must determine its magnitude and phase. Phase contains information on the evolution of kets, and is found from relativistic considerations. Magnitude will be determined from probability. It only makes sense to talk about probability when we are actually going to do a measurement. When we are actually going to do the measurement, a statement about hypothetical measurement, in the subjunctive mood, automatically becomes a statement about real measurement, in the future tense. This being the case, truth values for hypothetical results must become truth values for future events, i.e. probabilities. We require that the language generates the probability, , of getting the result, x, when a measurement of position is actually going to be performed on the state .

Two conditions are required of a probability density function». It must be greater than or equal to zero for all values of its argument, and it must have an area under its graph equal to 1 — or, in 3 dimensions, a hyper-volume under its graph of 1. Using the integral representation of the resolution of unity,
So, for a positive definite inner product, the function,
has the property of a probability density.


This definition ties down the relative magnitudes of the complex numbers, a and b, in rule II, and makes precise the weighting in logical OR.

Measurement

Since only a general principle has been used that it is possible to measure position, it is necessary to discuss other measurable quantities, or observables». I will be assume that all observables are a product of physical laws arising from interactions between particles. To describe a physically meaningful quantity, it will be required that an operator can be constructed from the operators describing the physical interactions of matter. A full analysis of a given measurement would require that the measurement apparatus as well as the system being measured be treated as an interacting multiparticle system. This must wait until a complete model of interactions between particles has been developed. Here general considerations are discussed, on the assumption that interactions will be described by linear operators and that a measurement is a physical process which can, at least in principle, be described as a combination of interaction operators.


This is a very broad definition of measurement, and does not require the intervention of a human observer. In classical physics, complete knowledge of the situation at a particular time contains sufficient information to determine the situation at any other time. When classical laws apply, and the state at some time is known from measurement, the state at all times can be calculated, at least in principle. For example, we can reasonably assume that Venus continues on its orbit while it is on the far side of the Sun and cannot be observed. Experience tells us that when Venus reappears it will be observed in precisely the position found by calculation from prior measurements. It is unreasonable to think that the same calculations do not hold while Venus is not observed. When classical laws apply, and the state has been measured at some time, then the state can be said to be effectively measured at any time, whether or not it is actually observed, and whether or not a calculation is actually performed.

Quantum Measurement

Measurement has two effects on the state of a particle, altering it due to the interaction of the apparatus with the particle, and also changing the information we have about the state. New information causes a change of state even in the absence of physical change because the state, or ket, is just a label for available information. The ket changes in part due to the effect of the apparatus on the particle, and in part due to the change in a conditional probability when the condition becomes known. This inverts the measurement problem»; collapse represents a change in information due to a new measurement but Schrödinger's equation» requires explanation — interference patterns are real. The requirement for a wave equation will be found in the next section, Time Evolution.

Classical probability theory describes situations in which every parameter exists, but some are not known. Probabilistic results come from different values taken by unknown parameters. We have a similar situation here, but now the unknowns are not describable as parameters. We assume no relationships between particles bar those generated by physical interaction. An experiment is described as a large configuration of particles incorporating the measuring apparatus as well as the process being measured. The configuration has been partially determined by setting up the experimental apparatus, reducing the possibilities to those with definite outcomes to the measurement, but the fine details of the configuration of matter is unknown. It is impossible, even in principle, to determine every detail of the configuration since the determination of each detail requires measurement, which in turn requires a larger apparatus containing new unknowns in the configuration of particles. Thus, there is always a lack of determination of initial conditions leading to randomness in the outcome, whether or not there is a fundamental indeterminism in Nature.

When we do a measurement, K, we get a definite result, a terminating decimal or n-tuple of terminating decimals read off the measurement apparatus. Let the possible results be k_i for i = 1, …, n . We assume that the dimension of H¹ is greater than n; this must be so if all measurements are reducible to measurements of position, and can be ensured through an appropriate choice of lattice at finer than the resolution of measurement. Each physical state is associated with a ket, labelled by the measurement result, so that if the measured result is k_i then the state is . The empirical determination of as a member of H¹ requires that we draw from experimental data the value of the inner product for an arbitrary state, . Without loss of generality and are normalised. By assumption, measurement of K is reducible to a set of measurements of position, so that each k_i is in one to one correspondence with the positions y_i of one or more particles used for the measurement (e.g. y_i may be the positions of one or more pointers). Then,
is the probability that a measurement of K has result k_i, given the initial state in H¹.

If the result is k_i, it is definitely k_i and cannot be given as k_j with i ≠ j by another measurement of K at the same time. So,
Measurement with result, k_i, implies a physical action on a system and is represented by the action of an operator, K_i, on Hilbert space. If a quantity is measurable we require that there is an element of physical reality associated with its measurement, meaning that, in measurement, the configuration of particles necessarily becomes such that the quantity has a well defined value. In practice, this means that, in the limit in which the time between two measurements goes to zero, a second measurement of the quantity necessarily gives the same result as the first. It follows that K_i is a projection operator», and has the form


The projection postulate was due originally to Von Neumann, for observables with discrete values. It was extended by Lüders to continuous observables. It is too restrictive to describe all numerical quantities used in the classical description of nature, and will be relaxed after a discussion of expectations.

Observable Operators

The expectation» of the result from a measurement of K, given an initial normalised state, in H¹, is


The effect of K on the state is to leave it unchanged.
k_i is the value of K in the state .


If a configuration of matter corresponds to an eigenstate» of an observable operator then the value of that observable exists independently of observation and is given by the corresponding eigenvalue». For physically meaningful observable quantities it will also be required that the Hermitian operator can be constructed, in some state, from the operators describing the physical interactions of matter. In a measurement of an observable, K, obeying the projection postulate, the probability that operators describing the interactions combine to is
Then can be understood as a classical probability function, where the random variable runs over the set of projection operators, K_i, corresponding to the outcomes of the measurement. The physical interpretation is that each K_i represents a set of unknown configurations of particle interactions in measurement, namely that set of configurations leading to the result k_i.

More generally, we have that if the state at time t is known to be , then the probability that a measurement at time t will give the result is given by

Classical Quantities

The projection postulate is required if the results of measurement are to be used to name states in Hilbert space, but classical quantities can also be defined from Hermitian operators when this is not the case. For an observable obeying the projection postulate, the corresponding measurement necessarily creates an eigenstate. More generally, a real value valued quantity is defined by the expectation of any hermitian operator
To say that an observable operator has a well defined value in a given state, a measurement should necessarily yield that value as the expectation of the operator. This is weaker than the projection postulate, which requires an eigenstate (in which the value is trivially given by the expectation).

Quantisation

In typical treatments of quantum mechanics based on canonical quantisation», 3-vector numerical quantities, x and p, representing position and momentum are simply replaced with operators x and p, and the canonical commutation relation, for a, b = 1, 2, 3,
is imposed. This procedure is usually justified only because it leads to an empirically accurate theory. In the present treatment, quantum theory has been set up from fundamental principles. It is necessary to define the operators x and p and to show that they obey the canonical commutation relation. The definitions follow the general from for observable operators.


The notation overloads the symbols x and p, in accordance with standard practice. Care should be taken over whether x and p are operators (as on the LHS) or numerical quantities (as on the RHS).

As an operator, the position observable is a map
So x maps the state with wave function to the state with wave function . In this treatment, wave functions are regarded as the coefficients of kets in a basis of position states. In some treatments wave functions are treated as a representation of Hilbert space. In this case states are identified with wave functions and the position observable is given by,


Proof:  For any plane wave state,
This is true for all states, , in a basis, so the theorem holds for all states by linearity.

As a mapping on the representation of wavefunctions,
and we write


Proof: For any ket ,
Hence,
So,
holds for all , as required.

The Uncertainty Principle

The uncertainty principle» can be derived from the theory while the theory cannot be derived from it. Strictly, this makes it a theorem, not a principle. Uncertainty in an observable is defined in the same way as variance» in statistics. It is a prediction of the root mean squared difference between observed values and the expected value».



where the commutator between operators A and B is [A, B] = AB − BA.

Proof:  Let and . Then, since A and B are Hermitian,
by the Cauchy-Schwarz inequality. Thus,
The general uncertainty relation follows by applying this result to and , observing that

The general uncertainty relation can be applied to any two observables. For example, the uncertainty in position-momentum is
In conventional units
This result, which is generally called the Heisenberg uncertainty principle, was first proved by Kennard, in 1927.

Time-Energy Uncertainty

Hilbert space has been set up to describe measurement at particular time. So time is a parameter, not an operator. The time-energy uncertainty relation. In the uncertainty relation for time-energy,
t does not refer to the time parameter. An intuitive interpretation is that a state which exists for only a short time, Δt, cannot have a definite energy, since to have a definite energy the frequency of the state needs to be defined over many cycles, the reciprocal of the required accuracy.

For example, in spectroscopy, excited states have a finite lifetime. By the time-energy uncertainty principle, they do not have a definite energy, and each time they decay the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the natural linewidth. Fast-decaying states have a broad linewidth, while slow decaying states have a narrow linewidth. Similarly, in particle physics, the faster a particle decays, the less certain is its mass.

In 1936, Dirac gave a precise definition and derivation of the time-energy uncertainty relation in a relativistic quantum theory of "events", in which particles follow trajectories parametrized independently by a different proper time. There is uncertainty in the time at which any event, or interaction for the particle, takes place. This “many-times” formulation inspired Shin-Ichiro Tomonaga»'s development of quantum electrodynamics.

Observables ↑ Evolution of Quantum States →