This is the important completeness relation,

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1 Observable quantities are represented by linear, hermitian operators! Compatible observables correspond to commuting operators! In addition to what was written in eqn 2.1.1, the vector corresponding to the state with value(s) a n of the of the observable A can be further expanded in terms of some other set of basis vectors: As long as the new basis corresponds to a complete set of basis vectors of a complete vector space. This is the important completeness relation, which is another way to state the universality of measurements in quantum mechanics. For a general state vector, we can then write

2 Note that for projection operators and with orthonormal subspaces: So, using this, what if we make a measurement of an observable A on a state which has been produced (by a general state collapsing to it) by another measurement of an observable B, which is incompatible (does not commute) with A. Say we have: So then, from the above definition of the projection operators in terms of eigenstates of a complete set, above, we would be able to write: So that the system is now left in a different state, multiplied by a complex scalar, as a result of the second measurement. Note that the very fact that this is the case

3 means that the two measurements are incompatible. Otherwise the seconds measurement would have had to return the same state, albeit with a different eigenvalue. The complex scalar coefficient is an indication that the original system (i.e. before collapse of the general state vector) had observable properties for both measurements, just that they were not compatible, with results stated above. The coefficient is the measure of the degree to which a sate that is known to have values corresponding to an observable B can also have certain values of the observable A. However, such a measure should be real and positive, so we use instead Which satisfies the Schwarz inequality (look it up for ordinary Euclidean space if you haven t seen it) for properly normalized states: We call this the probability: If we perform the same experiment many times on identically prepared systems, then is the probability p(α,β) that a system known to be in an eigenstate of A, with eigenvalue α can also be found in an eigenstate of B with eigenvalue β. I.e. the probability that such a system can measured values for both observables, even if they are not compatible. Of course, note that, if the two operators are compatible, then, for the case where both projection operators project out the same eigenvector: And for the case where the two projection operators project out different eigenvectors, then as required.

4 2.3 Interference Say we have an initial system which is prepared in such a way that half of it is in a state and the other half is in a state : The, if we make a general measurement on the system with an operator for which neither of these states is an eigenstate, then we get the following: As written above, the initial state is normalized to unity, and from the properties of the projection operators, we have Which is different from the result one would obtain if the measurement had first been done on one state in isolation and then the other. In that case we would have: In a sense, using the superposition of different states to describe our system is a statement of our ignorance of the system and in the context of a measurement means that the processes (which are encoded in the operators) giving rise to the two amplitudes and are indistinguishable. This gives rise to so called quantum mechanical interference with far reaching consequences such as measurable asymmetries in nature.

5 In the context of your first/second year double slit experiments with particles, the quantum mechanical interference can be interpreted to originate with the wave-like properties of particles. Probabilities do not simply add if the initial state involves a linear superposition of different states. In general, if a measurement (now literally what you do in the laboratory) can not distinguish between two processes (say the strong and weak interaction), then the probability of obtaining a particular measured value of an experimental observable must be consistent with the absolute value squared of the SUM of all amplitudes, corresponding to all possible inner products (meaning those that are allowed by so called quantum mechanical selection rules, some of which we will get to later) between the eigenstates of the operator(s) representing the observable. One more thing before we move on; suppose you have an initial state given by where η is a complex number of unit magnitude (a phase) : This still describes a valid normalized superposition of the states we used above, but if we make the same measurement now, we get With an interference term of Which means that the first measurement alone, isolating the superposition in the initial state from the general state vector is not enough to determine the relative phase between the two states. Another measurement is required to do so and it must be a measurement of some other incompatible observable.

6 2.4 Probability Amplitudes Consider a normalized ket vector in space; it represents some system in a state determined by some maximal set of compatible operators, as described above. The kets labeled The coefficients are defined by some other maximal set of operators. are the probability amplitudes corresponding to the probability of measuring the value if the system is known to be in the state. Definition: We call the probability amplitude for a system prepared in the state to be measured in the state. 2.5 Statistical Paramters The mean or average value The mean measured a value of an observable A for may identically prepared systems described by the vector is given in the usual statistical form, as the sum over all possible outcomes of the measurement times the probability that an eigenstate with the given eigenvalue is part of the system: The mean squared is similarly defined And in general, if one has an arbitrary function of the observed value we have The standard deviation (root-mean-square) From this we can obtain the standard deviation in the measures value in the usual way

7 Expectation Values The mean value of the measured value a of the observable A in the state is given by Since the operator A completely entails the properties of the observable and therefore all possible values of a, we can write and therefore say that the mean value or expectation value of the operator A in a given state is equal to the mean value of the corresponding variable or set of eigenvalues. 2.6 The General Uncertainty Relation Corresponding to the uncertainty in the measured value of an observable a from above, we introduce the uncertainty of an operator A, which are equal to each other:

8 Theorem: Theorem: The uncertainty A in a state is zero if and only if is an eigenstate of A. (Proof in Serot) If A and B are operators for two different observables, which may be incompatible, then in general, there are states for which ( A)( B) cannot be arbitrarily small, even though there are states for which A = 0 and other states for which B = 0. In fact (Proof in Serot) Comments: 1) If is an eigenstate of A but not at the same time of B is not, then (the operators are hermitian) The left hand side of the uncertainty relation is also zero due to the first theorem above (this page). Then the above uncertainty relation correctly reads 0 = 0. 2) If A and B are compatible, so that [A,B]=0, then we just get ( A) 2 ( B) 2 0 which carries little information, since one can always find states for which both B > 0 and A > 0. 3) If A and B are incompatible, so that [A,B] 0, then we get at least a lower bound on ( A) 2 ( B) 2. But the value of this bound depends on the particular state the system is in, which is arbitrary since the right hand side of the uncertainty relation vanishes for an eigenstate (i.e. we can make the bound as small as we want). 3) The most important case arises when we have [A,B] = αi, i.e. when the commutator is a (perhaps complex) number, since then, the right hand side of the uncertainty relation is state independent and if is properly normalized, which gives a fixed minimum uncertainty for any state in the vector space

9 We end up with a relation that should start to look familiar: So, non-commuting operators acting on general states that are not eigenstates of the operators give rise to the uncertainty relations which in turn (as we will see later) give rise to quantized values of the observables (i.e. quantize the system). 2.7 The Time Dependence of States We briefly deal here with the question of how states develop with time (you should have seen this before if not come talk to me): Time is not an observable (measurable) quantity in QM and therefore is not represented as an operator. Consider an arbitrary state at time t=0 which evolves into another state at time t : We can write this as a transformation as follows: For all where is a linear operator (just the transformation operator not an operator for a time observable!) 2.8 Conservation of Probability The state is not directly observable, but the probability amplitude (inner product) squared is proportional to the number of system described by that state and if no states are created or disappear during the time interval t of evolution, then So, we can write Then And the time evolution operator is unitary, only if probability is conserved!