The state vector and the Measurement Postulate- Book Excerpt #20
When Reality and Math Meet - Part II
This is section III.5 of the first volume of my book, “Quantum Physics: An Overview of a Weird World.” Here is the full table of contents and guidance on how to follow the book as it unfolds.
In the previous section I discussed the how a state vector represents the quantum state of a system, the Born rule, and the measurement postulate.
Note how the sum of all the probabilities adding up to unity is represented in Dirac’s notation neatly as:
If we want to deal with real-world measurements, we must specify which information from this mathematical construct (the wave function or state vector) we want to extract – that is, which physical quantity, and more precisely which ‘dynamical variable’, we want to measure (including, for example, position, momentum, angular momentum, energy, etc.). Mathematicians and physicists define an ‘operator’, which, as the word says, ‘operates’ on the wave function or state vector to obtain a specific piece of information associated with the observable.
Examples of mathematical operators that every one of us knows are the addition (+), subtraction (-), multiplication (*), and division (:) operators. The multiplication operator acts on two quantities, a and b, to obtain another quantity c as a*b=c. Similarly, one can conceive of an operator O acting on a state vector |Ψ⟩. If we consider this operation to be a formal equivalence to an act of measurement of a dynamical variable, the observable of a system, then the output of this operation should tell us something about the outcome of this measurement, a scalar quantity lambda, a number, which represents the expected value from this measurement, the so-called ‘eigenvalue’:
Lambda is the outcome of a measurement or, more broadly speaking, simply an ‘observation’. This equation is therefore called the ‘eigenequation’. It is the mathematical procedure we must perform to make predictions once we know the experimental setup and boundary conditions.
Stated a bit more precisely, one should say that each dynamical variable (e.g., position, momentum, angular momentum, energy, etc.) is associated with an (self-adjoint) operator O, the observable, which acts on the state vector of the quantum system |Ψ⟩ and which furnishes a scalar quantity, the measurement value, called the eigenvalue. Self-adjoint means that the complex conjugate of the operator is again the operator itself; however, these are mathematical aspects that are not very important for our purposes. Simply remember the connection between the classical notion of dynamical variables and its counterpart in QM as operators, the observables.
To avoid a possible source of confusion, let us state this again from another perspective. Operators are not numbers. 1,2,3,4,5… are numbers, whereas ‘+’, ‘-‘, ‘*’, or ‘:’ are binary operators. They are binary because they combine two numbers (called ‘operands’) to obtain another number (for example, 1+2=3). Therefore, it makes no sense to state that if O|Ψ⟩ = lambda*|Ψ⟩, then O = lambda, because you are then equating an operator with a number—that is, mistaking apples for bananas. Moreover, the operators used in QM operate on functions rather than on numbers. Perhaps the simplest non-trivial unary operator is the derivation operator:
It is unary because it operates on one function to obtain another function. Quantum operators—that is, observables—are also called ‘Qnumbers’ because they have a formal analogy with numbers. However, remember, they aren’t numbers at all.
Once that has become clear, here are some examples of observables. The simplest one is the position operator:
If we want to know the position of a particle, we multiply the wave function with a position variable x and then take the modulus square to obtain, as usual, the probability density in function of x. This tells us something about its whereabouts.
The momentum operator, that observable which tells us something about the momentum of the particle, has the form of a derivative operator.
Taking the derivative in space of the wave function (times the Planck constant bar and an imaginary number) will furnish a mathematical expression that tells us something about the speed (times mass) of the particle that the wave function describes.
Another important quantity in QM is the ‘expectation value’ of an operator – that is, the average value one obtains by measuring an observable many times. It is operatively a fundamental notion in QM because in the laboratory, what one usually must tackle in the real world are averaged measured values. For example, for the position operator X = x of a particle, one integrates its position x, weighting it by the probability of finding it in x or, in analytical terms (recall what we saw here):
It is the average value of the position x of a particle that one expects to obtain after a large number of measurements. More generally, this can be written in a more compact form by using Dirac’s notation, stating that the expectation value of an operator A in state Ψ is defined as:
with right-hand side Dirac bra-kets as defined in the integral above.
Another operator of paramount importance in QM is the energy operator, the ‘Hamiltonian’, which has a bit more complex structure. It is given by a second derivative in space (times a constant involving Planck’s constant and the mass of the particle), plus a function V(x), which represents the particle’s potential energy. By ‘potential energy‘, physicists mean the energy content of an object that can arise due to its position inside a force field (such as an object in a gravitational field or an electric charge in an electric force field) or stresses within itself (for example, an expanded spring) or in the form of chemical energy (the chemical bonds between atoms and molecules), etc. Without us getting into additional rigorous definitions and details, just imagine it as a sort of ‘stored energy’ (here, V(X) can be the function which tells us the stored energy an electron has at a certain distance x from the nucleus), which, however, can be transformed into another form of energy for practical purposes. The archetypal example is the transformation of potential gravitational energy (say, the potential energy of a stone at the top of a hill) into kinetic energy (the stone rolling down the hill and acquiring a certain speed.) The Hamiltonian operator is written as follows:
It is this latter observable in particular which leads to the famous ‘time-independent Schrödinger equation’:
This is, however, not the most general equation because, as the name indicates, it is time-independent, while physical systems are, in many cases, time-dependent. Therefore, the time-dependent energy operator is called ‘evolution operator’ and is given by:
That’s why you will frequently find the ‘time-dependent Schrödinger equation’ in this form:
An important property of Schrödinger’s equation is that it is linear. A differential equation is said to be ‘linear’ when, given two different allowed solutions, their sum is also an allowed solution. As an example, you might recall the superposition of two waves forming a new wave, as discussed with the interference phenomena. This will be crucial later, when we discuss the quantum superposition principle.
The Hamiltonian operator also defines the ‘time evolution’ of the wave function from a time 0 to a time t. In particular, if H is independent of time, given a state at an initial time t=0, the state of the quantum system at any subsequent time t is ‘generated’ by the so-called ‘unitary operator’
as
Its main physical interpretation is that it is an operator that does not describe an observable (that is, a measurement), but the evolution in time of a quantum system. It changes the phase of the wave function or state vector but not its length. This ‘length conservation’ reflects nothing else than the necessity to conserve the sum of all the probabilities associated to the eigenstates before and after the state evolution, that is, it must hold:
Therefore, its name ‘unitary operator’. Moreover, note that if U is the unitary operator that evolves the quantum state forward in time, its complex conjugate
sends it backwards in time by an interval t. Then it becomes obvious that applying first and (or vice versa) consecutively it preserves the state, that is, an equivalent formal definition for a unitary operator is that it satisfies the general relation:
With one being a scalar (just the number ‘1’) or, more generally, an identity operator. This reflects a fundamental property of QM, that of ‘unitarity’, which must always hold.
This concludes the series of excerpts from my book. If you would like to browse selected excerpts from the two volumes in digital flipbook format, you can find them here and here.
You can order the books through Amazon here and here. If Amazon is not your cup of tea, you can also get the ebooks (EPUB or PDF) directly from me in exchange for a number of coffees.
