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Chapter 5
Probability, Expectation Values,
and Uncertainties
As indicated earlier, one of the remarkable features of the physical world is that randomness is incarnate, irreducible. This is mirrored in quantum theory by the appearance of a quantity, the wave function, which gives us probabilistic information about the properties of a physical system. As a consequence, there enters into quantum theory many of the concepts that had their origins in the analysis of classical probabilitistic processes, such as the average (or expectation) value of some random quantity, and its standard deviation, or uncertainty. These concepts are defined in exactly the same way in the quantum theory as in classical theory of probability and statistics; it is just the origin of the probability in the first place that is unusual. In this chapter we will discuss some of these concepts in the context of wave mechanics.
5.1 The Probability Interpretation of the Wave Function
For the particular case of measuring the position of a particle, we have seen a kind of behaviour that is found to occur throughout the physical world: if we repeat an experiment under exactly identical conditions, i is almost always found that the outcome of any measurement that we perform will vary in a random way, and this randomness is not due to any flaw in the way we have conducted the experiment, it is intrinsic to all physical systems – it is a law of nature. We have seen this explicitly only in the case of the two slit experiment wherein electrons are each prepared in exactly the same way – i.e. exactly the same experimental procedures are applied to each electron before being sent on its journey through the two slit apparatus – and yet the electrons are observed to arrive at the observation screen at points that vary in a random way each time.
The fact that interference effects were observed in the two slit experiment then lead us to propose that associated with each identically prepared electron is a wave functionΨ(x, t), which we are further lead to believe carries with it all the information that can be possibly obtained about the electron, such as its position and momentum. More than that, the fact that this inormation is encoded into a wave means that there are limitations on this information, as embodied in the Heisenberg uncertainty principle. Thus we are lead to think of the wave function itself as representing the state of the system, in contrast to what we mean by state according to classical physics which is a list, in principle, of the values of every physical property at some instant in time, to unlimited precision.
The randomness in the measurable position exhibited by a single particle is then quantified via the probability interpretation of the wave function, as introduced in the preceding Chapter. It is restated here for convenience: if a particle is described by a wave function Ψ(x, t), then |Ψ(x, t)|^2 δx = Probability of observing the particle in the region (x, x + δx) at time t.
It is important to understand what this statement means. What is implied here is that if we were to repeat the experiment of measuring the position of a single particle many times over under identical conditions, a fraction |Ψ(x, t)|^2 δx of the experiments will give a result lying in the range (x, x + δx). Usually, in talking about this interpretation, use is made of the notion of an ‘ensemble of identically prepared systems’. By this we mean we imagine that we set up an experimental apparatus and use it to prepare an extremely large number N of particles all in the same state. We then propose to measure the position of each particle at some time t after the start of the preparation procedure. Because each particle has gone through exactly the same preparation, they will all be in the same state at time t, as given by the wave function Ψ(x, t). The collection of particles is known as an ensemble, and it is common practice (and a point of contention) in quantum mechanics to refer to the wave function as the wave function of the ensemble, rather than the wave function of each individual particle. We will however often refer to the wave function as if it is associated with a single particle, in part because this reflects the more recent development of the point-of-view that the wave function represents the information that we have about a given particle (or system, in general) 1.
So suppose we have this ensemble of particles all in the state Ψ(x, t), and we measure the position of each particle at time t. In general, we will get a scatter of results, so suppose we divide the range of x values into regions of width δx, and measure the number of times that we measure the position x of the particle to lie in the range (x, x + δx), δN (x) say. The fraction of times that the particle is observed to lie in this range will then be
δN (x, t) N
≈ P (x, t)δx (5.1)
As the number N of measurements becomes larger, this ratio will approach the probability given by P (x, t) = |Ψ(x, t)|^2 i.e.
lim N →∞
δN (x, t) N δx = P (x, t). (5.2)
(^1) In that case, the probability is then the probability in the sense of giving the odds for various horses to win a particular race: there is no ‘ensemble’ of horse races on the basis of which to calculate the odds, after all, the race is run only once – but there is information drawn from elsewhere from which, by an algorithm known only to bookmakers, these odds can be assigned. Here, the information we know, the wave function, gives us probabilities by the algorithm known as the Born interpretation.
Finally we note that provided the wave function is normalized to unity, the probability of finding the particle over a finite region of space, say a < x < b, will be given by
∫ (^) b
a
|Ψ(x, t)|^2 dx. (5.5)
5.2 Expectation Values and Uncertainties
Given that the wave function gives the probability distribution of the random data gen- erated by repeating the same experiment (measuring the position of a particle) under identical conditions, we can use this probability distribution to calculate a number quan- tities that typically arise in the analysis of random data: the mean value and standard deviation.
So suppose once again that we have this ensemble of particles all in the state Ψ(x, t), and we measure the position of each particle at time t. The fraction of particles that are observed to lie in this range will then be
δN (x, t) N
≈ P (x, t)δx (5.6)
where the approximate equality will become more exact as the number of particles becomes larger. We can then calculate the average value of all these results in the usual way, i.e.
〈x(t)〉 =
All δx
x δN (x, t) N
All δx
x P (x, t)δx. (5.7)
In the limit in which the number of particles becomes infinitely large, and as δx → 0, this becomes an integral:
〈x(t)〉 =
−∞
x P (x, t) dx =
−∞
x |Ψ(x, t)|^2 dx (5.8)
i.e. this is the average value of, in effect, an infinite number of measurements of the position of the particles all prepared in the same state described by the wave function Ψ(x, t). It is usually referred to as the expectation value of x. Similarly, expectation values of functions of x can be derived. For f (x), we have
〈f (x)〉 =
−∞
f (x) |Ψ(x, t)|^2 dx. (5.9)
In particular, we have
〈x^2 〉 =
−∞
x^2 |Ψ(x, t)|^2 dx. (5.10)
We can now define the uncertainty in the position of the particle by the usual statistical quantity, the standard deviation (∆x)^2 , given by
(∆x)^2 = 〈x^2 〉 − 〈x〉^2. (5.11)
Of course, it is not only the position of the particle that can be determined (at least in probabilistic fashion) from the wave function. If we are to believe that the wave function is the repository of all information about the properties of a particle, we ought to be able to say something about the properties of the velocity of the particle. To this end, we introduce the average velocity into the picture in the following way.
We have shown above that the expectation value of the position of a particle described by a wave function Ψ(x, t) is given by Eq. (5.8). It is therefore reasonable to presume that the average value of the velocity is just the derivative of this expression, i.e.
〈v(t)〉 = d dt
−∞
x |Ψ(x, t)|^2 dx
−∞
x
∂t |Ψ(x, t)|^2 dx
−∞
x
[
∂Ψ∗^ (x, t) ∂t Ψ(x, t) + Ψ∗^ (x, t)
∂Ψ(x, t) ∂t
]
dx. (5.12)
More usually, it is the average value of the momentum that is of deeper significance than that of velocity. So we rewrite this last result as the average of the momentum. We also note the the term in [... ] can be written as the real part of a complex quantity, so that we end up with
〈p〉 = 2mRe
[∫ +∞
−∞
x Ψ∗^ (x, t) ∂Ψ(x, t) ∂t
dx
]
Later we will see that there is a far more generally useful (and fundamental) expression for the expectation value of momentum, which also allows us to define the uncertainty ∆p in momentum. For the present, in order to illustrate the sort of results that can be obtained from the wave function, we will consider a particularly simple example about which much can be said, even with the limited understanding that we have at this stage. The model is that of a particle in an infinitely deep potential well.
5.3 Particle in an Infinite Potential Well
Suppose we have a single particle of mass m confined to within a region 0 < x < L with potential energy V = 0 bounded by infinitely high potential barriers, i.e. V = ∞ for x < 0 and x > L. This simple model is sufficient to describe (in one dimension), for instance, the properties of the conduction electrons in a metal (in the so-called free electron model), or the properties of gas particles in an ideal gas where the particles do not interact with each other. We want to learn as much about the properties of the particle using what we have learned about the wave function above.
The first point to note is that, because of the infinitely high barriers, the particle cannot be found in the regions x > L and x < 0. Consequently, the wave function has to be zero in these regions. If we make the not unreasonable assumption that the wave function has to be continuous, then we must conclude that
Ψ(0, t) = Ψ(L, t) = 0. (5.14)
These conditions on Ψ(x, t) are known as boundary conditions.
Between the barriers, the energy of the particle is purely kinetic. Suppose the energy of the particle is E, so that
E = p 2 2 m
Using the de Broglie relation E = ℏk we then have that
k = ±
2 mE ℏ
We can now check on the other boundary condition, i.e. that Ψ(L, t) = 0, which leads to:
sin(kL) = 0 (5.24)
and hence kL = nπ n an integer (5.25)
which implies that k can have only a restricted set of values given by
kn =
nπ L
An immediate consequence of this is that the energy of the particle is limited to the values
E (^) n =
ℏ 2 k (^2) n 2 m
π 2 n 2 ℏ 2 2 mL^2 = ℏω (^) n (5.27)
i.e. the energy is ‘quantized’.
Using these values of k in the normalization condition leads to
∫ (^) +∞
−∞
|Ψ(x, t)|^2 dx = 4|A|^2
∫ L
0
sin^2 (kn x) = 2|A|^2 L (5.28)
so that by making the choice
A =
2 L
eiφ^ (5.29)
where φ is an unknown phase factor, we ensure that the wave function is indeed normalized to unity. Nothing we have seen above can give us a value for φ, but whatever choice is made, it always found to cancel out in any calculation of a physically observable result, so its value can be set to suit our convenience. Here, we will choose φ = −π/2 and hence
A = −i
2 L
The wave function therefore becomes
Ψn (x, t) =
L
sin(nπx/L)e−iωn^ t^0 < x < L
= 0 x < 0 , x > L. (5.31)
with associated energies
E (^) n =
π 2 n 2 ℏ 2 2 mL^2 n = 1, 2 , 3 ,.... (5.32)
where the wave function and the energies has been labelled by the quantity n, known as a quantum number. It can have the values n = 1, 2 , 3 ,... , i.e. n = 0 is excluded, for then the wave function vanishes everywhere, and also excluded are the negative integers since they yield the same set of wave functions, and the same energies.
We see that the particle can only have the energies E (^) n , and in particular, the lowest energy, E 1 is greater than zero, as is required by the uncertainty principle. Thus the energy of the particle is quantized, in contrast to the classical situation in which the particle can have any energy ≥ 0.
5.3.1 Some Properties of Infinite Well Wave Functions
The wave functions derived above define the probability distributions for finding the par- ticle of a given energy in some region in space. But the wave functions also possess other important properties, some of them of a purely mathematical nature that prove to be extremely important in further development of the theory, but also providing other information about the physical properties of the particle.
Energy Eigenvalues and Eigenfunctions
The above wave functions can be written in the form
Ψn (x, t) = ψ (^) n (x)e−iEn^ t/ℏ^ (5.33)
where we note the time dependence factors out of the overall wave function as a complex exponential of the form exp[−iEn t/ℏ]. As will be seen later, the time dependence of the wave function for any system in a state of given energy is always of this form.
The energy of the particle is limited to the values specified in Eq. (5.32). This phenomenon of energy quantization is to be found in all systems in which a particle is confined by an attractive potential such as the Coulomb potential binding an electron to a proton in the hydrogen atom, or the attractive potential of a simple harmonic oscillator. In all cases, the boundary condition that the wave function vanish at infinity guarantees that only a discrete set of wave functions are possible, and each is associated with a certain energy – hence the energy levels of the hydrogen atom, for instance.
The remaining factor ψ (^) n (x) contains all the spatial dependence of the wave function. We also note a ‘pairing’ of the wave function ψ (^) n (x) with the allowed energy E (^) n. The wave function ψ (^) n (x) is known as an energy eigenfunction and the associated energy is known as the energy eigenvalue. This terminology has its origins in the more general formulation of quantum mechanics we will be considering in later Chapters.
Probability Distributions
The probability distributions corresponding to the wave functions obtained above are
Pn (x) = |Ψn (x, t)|^2 =
L
sin^2 (nπx/L) 0 < x < L = 0 x < 0 , x > L
(5.34) which are all independent of time, i.e. these are analogous to the stationary states of the hydrogen atom introduced by Bohr – states whose properties do not change in time. The nomenclature stationary state is retained in modern quantum mechanics for such states.
We can plot P (^) n (x) as a function of x for various values of n to see what we can learn about the properties of the particle in the well (see Fig. (5.2)). We note that P (^) n is not uniform across the well. In fact, there are regions where it is very unlikely to observe the particle, whereas elsewhere the chances are maximized. If n becomes very large (see Fig. (5.2)(d)), the probability oscillates very rapidly, averaging out to be 1/L, so that the particle is equally likely to be found anywhere in the well. This is what would be found classically if the particle were simply bouncing back and forth between the walls of the
Orthonormality
An important feature of the wave functions derived above follows from considering the following integral: ∫ (^) +∞
−∞
ψ ∗ m (x)ψ (^) n (x) dx =
L
∫ L
0
sin(mπx/L) sin(nπx/L) dx = δ (^) mn (5.39)
where δ (^) mn is known as the Kronecker delta, and has the property that
δ (^) mn = 1 m = n = 0 m )= n
Thus, if m = n, the integral is unity, as it should be as this is just the normalization condi- tion that was imposed earlier on the wave functions. However, if m )= n, then the integral vanishes. The two wave functions are then said to be orthogonal, a condition somewhat analogous to two vectors being orthogonal. The functions ψ (^) n for all n = 1, 2 , 3 ,... are then said to be orthonormal.
The property of orthonormality of eigenfunctions is found to be a general property of the states of quantum systems that will be further explored in later Chapters.
Linear Superpositions
We found earlier that if we combine together wave functions of different wavelength and frequency, corresponding to different particle momenta and energies, we produce something that had an acceptable physical meaning – a wave packet. What we will do now is combine together – but in this case only two – different wave functions and see what meaning can be given to the result obtained. Thus, we will consider the following linear combination, or linear superposition, of two wave functions:
Ψ(x, t) =
[
Ψ 1 (x, t) + Ψ 2 (x, t)
]
L
[
sin(πx/L)e−iω^1 t^ + sin(2πx/L)e−iω^2 t^
]
0 < x < L
= 0 x < 0 and x > L. (5.41)
The factor 1/
2 guarantees that the wave function is normalized to unity, as can be seen by calculating the normalization integral Eq. (5.3) for the wave function defined in Eq. (5.41).
How are we to interpret this wave function? Superficially, it is seen to be made up of two wave functions associated with the particle having energies E 1 and E 2. These wave functions contribute equally to the total wave function Ψ(x, t) in the sense that they both have the same amplitude, so it is tempting to believe that if we were to measure the energy of the particle rather than its position, we would get either result E 1 and E 2 with equal probability of 12. This interpretation in fact turns out to be the case as we will see later. But the fact that the particle does not have a definite energy has important consequences as can be seen by considering the probability distribution for the position of the particle.
This probability distribution is
P (x, t) = |Ψ(x, t)|^2
=
L
[
sin^2 (πx/L) + sin^2 (2πx/L) + 2 sin(πx/L) sin(2πx/L) cos(∆ωt)
] (5.42)
where ∆ω = (E 2 − E 1 )/ℏ. This is obviously a time dependent probability distribution, in contrast to what was found for the eigenfunctions Ψn (x, t). In other words, if the wave function is made up of contributions of different energies, the particle is not in a stationary state.
In Fig. (5.3), this probability distribution is plotted at three times. At t = 0, the proba- bility distribution is
P (x, 0) =
L
(sin(πx/L) + sin(2πx/L))^2 (5.43)
which results in the distribution being peaked on the left hand side of the well. At the time t = π/ 2 ∆ω, the time dependent term vanishes and the distribution is
P (x, π/ 2 ∆ω) =
L
(sin^2 (πx/L) + sin^2 (2πx/L)). (5.44)
Finally, at time t = π/∆ω, the distribution is
P (x, π/∆ω) =
L
(sin(πx/L) − sin(2πx/L))^2 (5.45)
which gives a peak on the right hand side of the well. Thus, the maximum probability swings from the left to the right hand side of the well (and back again), i.e. there is increased probability of finding the particle particle first on one side of the well, and then at a time π/∆ω later on the other side but without the maximum moving through the centre of the well. This is counterintuitive: the maximum would be expected to also move back and forth between the walls, mirroring the expected classical behaviour of the particle bouncing back and forth between the walls.
(a) (b) (c)
P (x, 0) P (x, π/ 2 ∆ω) P (x, π/∆ω)
x x x
Figure 5.3: Plots of the time dependent probability distributions P (x, t) = (Ψ 1 (x, t)+Ψ 2 (x, t))/
√ 2 for a particle in an infinite potential well. The times are (a) t = 0, (b) t = π/ 2 ∆ω and (c) t = π/∆ω where ℏ∆ω = E 2 − E 1.
As a final illustration, we can find the total probability of finding the particle on the left hand half of the well, i.e. in the region 0 < x < L/2:
PL =
∫ L/ 2
0
P (x, t) dx = 12 + π −^1 cos(∆ωt) (5.46)
while the corresponding result for the right hand side is
P (^) R = 12 − π −^1 cos(∆ωt) (5.47)
which perhaps better illustrates the ‘see-sawing’ of the probability from one side to the other with a frequency of 2π∆ω.
or, in other words ∆x∆p ≥ 12 ℏ (5.54)
in agreement with the Heisenberg uncertainty principle, Eq. (3.14).
