As you may have observed in working through earlier chapters, some of the concepts and mathematical techniques of classical mechanics may be extended to the domain of quantum mechanics. But the fundamentally probabilistic nature of quantum mechanics leads to several profound differences, and it’s very important for you to develop a firm grasp of those differences. That grasp includes an understanding of how certain classical-physics terminology does or does not apply to quantum mechanics.
Fortunately, progress has been made in developing consistent terminology in the roughly 100 years since the birth of quantum mechanics, but if you read the most popular quantum texts and online resources, you’re likely to notice some variation in the use of the terms “quantum state” and “wavefunc- tion.” Although some authors use these terms interchangeably, others draw a significant distinction between them, and that distinction is explained in this section.
In the most common use of the term, the quantum state of a particle or system is a description that contains all the information that can be known about the particle or system. A quantum state is usually written asψ (sometimes uppercase, especially when time dependence is included) and can be represented by a basis-independent ket|ψor|. Quantum states are members of an abstract vector space and obey the rules for such spaces, and the Schr¨odinger equation describes how a quantum state evolves over time.
So what’s the difference between a quantum state and a quantum wavefunc- tion? In a number of quantum texts, a quantum wavefunction is defined as the expansion of a quantum state in a specified basis. And which basis is that?
Whichever basis you choose, and a logical choice is the basis corresponding to the observable of interest. Recall that every observable is associated with an operator, and the eigenfunctions of that operator form a complete orthogonal basis. That means that any function may be synthesized by weighted combi- nation (superposition) of those eigenfunctions. As described inSection 1.6, if you expand the quantum state using a weighted sum of the eigenfunctions
(ψ1,ψ2,. . ., ψN) for that basis, then a state represented by ket|ψ may be written as
|ψ =c1|ψ1 +c2|ψ2 + · · · +cN|ψN = N n=1
cn|ψn, (1.35) and the wavefunction is the amount (cn) of each eigenfunction|ψn in state
|ψ. So the wavefunction in a specified basis is the collection of (potentially complex) valuescnfor that basis.
Also as described inSection 1.6, eachcnmay be found by projecting state
|ψonto the corresponding (normalized) eigenfunction|ψn:
cn= ψn|ψ. (4.1)
The possible measurement outcomes are the eigenvalues of the operator corresponding to the observable, and the probability of each outcome is proportional to the square of the magnitude of the wavefunction valuecn. Thus the wavefunction represents the “probability amplitude” of each outcome.1
If it seems strange to apply the word “function” to a group of discrete values cn, the reason for that terminology should become clear when you consider a quantum system (such as afree particle) in which the possible outcomes of measurements (the eigenvalues of the operator associated with the observable) are continuous functions rather than discrete values. In quantum textbooks, this is sometimes described as the operator having a “continuous spectrum” of eigenvalues.
In that case, the matrix representing the operator associated with an observable, such as position or momentum, has an infinite number of rows and columns, and there exist an infinite number of eigenfunctions for that observable. For example, the (one-dimensional) position basis functions may be represented by the ket|x, so expanding the state represented by ket|ψin the position basis looks like this:
|ψ = ∞
−∞ψ(x)|xdx. (4.2)
Notice that the “amount” of the basis function|xat each value of the continu- ous variablexis now the continuous functionψ(x). So the wavefunction in this case is not a collection of discrete values (such ascn), but rather the continuous function of positionψ(x).
1The word “amplitude” is used in analogy with other types of waves, for which the intensity is proportional to the square of the wave’s amplitude.
To determineψ(x), do exactly as you do in the discrete case: project the state|ψonto the position basis functions:
ψ(x)= x|ψ. (4.3)
Just as in the discrete case, the probability of each outcome is related to the square of the wavefunction. But in the continuous case|ψ(x)|2gives you the probability density (the probability per unit length in the 1-D case), which you must integrate over a range ofxto determine the probability of an outcome within that range.
The same approach can be taken for the momentum wavefunction. The (one-dimensional) momentum basis functions may be represented by the ket
|p, and expanding the state|ψin the momentum basis looks like this:
|ψ = ∞
−∞φ(p)˜ |pdp. (4.4)
In this case the “amount” of the basis function at each value of the continuous variablepis the continuous functionφ(p).˜
To determine φ(p), project the state˜ |ψ onto the momentum basis functions:
φ(p)˜ = p|ψ. (4.5)
So for a given quantum state represented by|ψ, the proper approach to finding the wavefunction in a specified basis is to use the inner product to project the quantum state onto the eigenfunctions for that basis. But research studies2 have shown that even after completing an introductory course on quantum mechanics, many students are unclear on the relationship of quantum states, wavefunctions, and operators.
One common misconception concerning quantum operators is that if you’re given a quantum state|ψ, you can determine the position-basis wavefunction ψ(x)or the momentum-basis wavefunction φ(p)˜ by operating on state |ψ with the position or momentum operator. This is not true; as described previously, the correct way to determine the position or momentum waveform is to project the state|ψonto the eigenstates of position or momentum using the inner product.
A related misconception is that an operator may be used to convert between the position-basis wavefunctionψ(x)and the momentum-basis wavefunction φ(p). But as you’ll see in˜ Section 4.4, the position-basis and momentum-basis
2See, for example, [4].
wavefunctions are related to one another by theFourier transform, not by using the position or momentum operator.
It’s also common for students new to quantum mechanics to believe that applying an operator to a quantum state is the analytical equivalent to making a physical measurement of the observable associated with that operator. Such confusion is understandable, since operating on a state does produce a new state, and many students have heard that making a measurement causes the collapse of a quantum wavefunction. The actual relationship between applying an operator and making a measurement is a bit more complex, but also more informative. The measurement of an observable does indeed cause a quantum state to collapse to one of the eigenstates of the operator associated with that observable (unless the state is already an eigenstate of that operator), but that is definitely not what happens when you apply an operator to a quantum state.
Instead, applying the operator produces a new quantum state that is the superposition (that is, the weighted combination) of the eigenstates of that operator. In that superposition of eigenstates, the weighting coefficient of each eigenstate is not just the “amount” (cn) of that eigenstate, as it was in the expression for the state pre-operation:
|ψ =ncn|ψn.
But after applying the operator to the quantum state, the weighting factor for each eigenstate includes the eigenvalue of that eigenstate, because the operator has brought out a factor of that eigenvalue (on):
O|ψ =ncnO|ψn =ncnon|ψn. (4.6) As explained in Section 2.5, forming the inner product of this new state O|ψwith the original state|ψgives the expectation value of the observable corresponding to the operator. So in the case of observableOwith associated operatorO, eigenvaluesonand expectation valueO,
ψ|O|ψ =m(c∗mψm|)n(cnon|ψn)
=mn(c∗moncn)ψm|ψn =non(|cn|)2= O,
sinceψm|ψn = δm,n for orthonormal wavefunctions. Note the role of the operator: it has performed the function of producing a new state in which the weighting coefficient of each eigenfunction has been multiplied by the eigenvalue of that eigenfunction. This is a crucial step in determining the expectation value of an observable.
The bottom line is this: applying an operator to the quantum state of a system changes that state by multiplying each constituent eigenfunction by its eigenvalue (Eq. 4.6), and making a measurement changes the quantum state by causing the wavefunction to collapse to one of those eigenfunctions.
So operators and measurements both change the quantum state of a system, but not in the same way.
You can see examples of quantum operators in action later in this chapter and inChapter 5, but before getting to that, you may find it helpful to consider the general characteristics of quantum wavefunctions, which is the subject of the next section.