195a Physics Course Notes Ideas of Quantum Mechanics 021024 F. Porter
1
Introduction
This note summarizes and examines the foundations of quantum mechanics, including the mathematical background.
2
2.1
General Review of the Ideas of Quantum Mechanics
States
We have in mind that there is a system, which is describable in terms of possible states. A system could be something simple, such as a single electron, or complex, such as a table. Suppose we have a system consisting of N spinless particles. We use the term particle to denote any object for which any internal structure is unimportant. Classically, we may describe the state of this system by specifying, at some time t the generalized coordinates and momenta: {qi (t), pi (t), i = 1, 2, . . . , N } , (1)
where the spatial dimensionality of the qi and pi is implicit. The time evolution of this system is given by Hamiltons equations: qi = pi H pi H = qi (2) (3)
In quantum mechanics, it is not possible to give such a complete specication to arbitrary precision. For example, the limit to how well we may specify the position and momentum of a particle in one dimension is limited by the uncertainty principle: xp 1/2, where indicates a range of possible values. Well investigate this relation more explicitly later, but for 1
now it should just be a reminder of your elementary quantum mechanics understanding. We must be content with selecting a suitable set of quantities which can be simultaneously specied to describe the state. We refer to this set as a Complete Set of Commuting Observables (CSCO). Specifying a CSCO corresponds to specifying the eigenvalues of an appropriate complete set of commuting Hermitian operators, for the state in question. Measurements (eigenvalues of Hermitian operators) of other quantities cannot be predicted with certainty, only probabilities of outcomes can be given. The evolution in time of the system is described by a wave equation, for example, the Schrdinger equation. o
2.2
Probability Amplitudes
The quantum mechanical state of a system is described in terms of waves, called probability amplitudes, or just amplitudes for short. Note that probabilities themselves are always non-negative, so it is more dicult to imagine the probabilities themselves as wavelike. Instead, the probabilities are obtained by squaring the amplitudes: Probability | |2 , (4)
where stands for the amplitude. More explicitly, the probability of observing state variable (e.g., position) x in volume element d3 (x) around x is equal to: (5) | (x)|2 d3 (x). A quantum mechanical probability is analogous to the intensity of a classical wave. The quantum mechanical wave evolves in time according to a time evolution operator, eiHt involving the Hamiltonian, H . Hence, if eiHt 0 (x) = (x, t), (6)
where 0 (x) is the wave function at t = 0 in terms of coordinate position x, then dierentiation gives: i d (x, t) = H (x, t). dt (7)
We recognize this as the Schrdinger equation. Thus, the temporal frequency o of the wave is determined by the energy structure. For a particle of energy 2
E , the frequency is = E (or = E/2 ). This hypothesis is also applied in relativistic situations, for example, for a photon. The spatial behavior of a wave is given by the deBroglie hypothesis: A particle is described as a quantum mechanical wave with wavelength: 1 =, 2 p (8)
or with wavenumber k = 1/ = p. This relation is assumed to also hold relativistically. We may make a brief aside on the subject of dispersion relations. As in classical electrodynamics, the relation between and k for a wave is called a dispersion relation. In the case of the quantum mechanics of a free particle of mass m, the dispersion relation is = k 2 /2m (9)
for a non-relativistic particle (when we do not include the rest mass in the energy, hence E = p2 /2m), and 2 = k 2 + m2 for a relativistic particle. (10)
2.3
Wave Equations
In quantum mechanics, the dynamics is determined by the wave equation. The form of the wave equation is given by the dispersion relation. By analogy with light, and ignoring issues of mathematical rigor, let us build physical waves describing a particle of mass m from superpositions of plane waves. Note well that we are assuming that the wave equation is linear, so linear combinations of solutions are also solutions. Our plane wave building blocks are: (x, t) = Aei(kxt) , (11) where k=p and = E . We presume that this forms a complete set of functions, that is, any physical state may be expanded as a linear superpostion of elements of this set. Let us suppose we have a free particle. We search for a dierential equation which is satised by all plane waves which could
3
describe the particle. We shall postulate this to be our wave equation. As we saw above, the dispersion equation for a (possibly relativistic) free particle is E 2 p2 = m2 , or 2 k 2 = m2 . 2 (x, t) = 2 (x, t), 2 t 2 (x, t) = k 2 (x, t), we have (12)
(13)
Considering the following derivatives of the plane waves (14) (15)
2 2 = m2 . (16) 2 t We postulate this to be the desired wave equation. It is known as the KleinGordon equation. It describes the motion, in quantum mechanics, of a free particle of mass m. However, free particles quickly become boring; we really want to be able to discuss interactions, e.g., interacting particles. Demanding relativistic invariance leads us into quantum eld theory. However, we often dont require full invariance, and typically make two very useful simplifying assumptions in non-relativistic quantum mechanics: The creation and destruction of particles is assumed not to occur. The number of each particle type is constant. However, there is occasional need to make exceptions to this assumption; the most notable is for the photon. All particles (again, except for photons!) are assumed to be nonrelativistic. Typically this means we stop at order v 2 in the energy, but sometimes we carry out calculations to higher order. We have already seen that these assumptions are reasonable for ordinary atomic systems. In non-relativistic quantum mechanics (Schrdinger theory) the wave o function (x, t) really has the precise meaning:
4
The probability that a measurement of the particles position at time t will yield x in d3 (x) around x is: | (x, t)|2d3 (x). | (x, t)|2 is a probability density. Note that (x, t) and (x, t) = ei (x, t), where is a real number, describe the same physical situation, since the probability density is unchanged [where we make the inherent assumption that it is probabilities we can measure, not probability amplitudes]. Let us take the non-relativistic limit of our free particle wave equation: E= m2 + p2 p2 + pO 2m p m
3
(17)
= m+ Hence,
.
(18)
(x, t) = Aei(pxEt) Aei(px 2m t) eimt S (x, t)eimt , where S (x, t) ei(px 2m t) .
p2 p2
(19)
(20)
But | (x, t)|2 = |S (x, t)|2 , so we may drop the overall eimt phase factor and look for a linear dierential equation satised by our non-relativistic plane wave solutions (dropping the S subscript now). We have p2 = i , t 2m 2 = p2 . Thus, 12 = . (23) t 2m This is the Schrdinger equation for a free particle of mass m. Note the o correspondence with the dispersion relation, Eq. 9. i 5 (21) (22)
In the non-relativistic case, it is easy to generalize to situations where the particle is not free: Introduce a potential function, V (x, t) to describe interactions. Our hypothesis is that the time dependence of the wave is determined by = E . With V = 0, this gives E =T +V = Thus, i = i exp [i(p x Et)] t t = E = H, p2 + V (x, t). 2m (24)
(25)
where H = T + V is the Hamiltionian operator. We may sometimes need to make the distinction between an operator and its spectral values (eigenvalues) more explicit. When this arises, we will use notation of the form H or Hop to denote the operator. However, we usually rely on context, and omit such notational guides. We also have that p2 12 (x, t) = (x, t) = (E V ) (x, t). 2m 2m (26)
Putting this together with Eq, 25, we nd: H (x, t) = E (x, t) it (x, t) = 12 (x, t) + V (x, t) (x, t). 2m (27)
The upper form, in which E is the energy eignenvalue for a static potential, is referred to as the time-independent Schrdinger equation. The lower o form is referred to as the time-dependent Schrdinger equation. o
3
Mathematical Considerations
Let us take a step back now, and set up a more rigorous mathematical framework in which to implement the notions we have been discussing. It is a highly reassuring feature of quantum mechanics that we are able to do so. We have decided to describe particles by waves, giving probability amplitudes, where absolute squares lead to measurable physical probabilities. 6
Waves are conveniently described by complex-valued functions of whatever generalized coordinates are involved. An essential feature is that waves interfere, hence our state space must allow for the possibility of superposition of waves. The requirement of superposability suggests that the state space of permissible wave functions should be a vector space. This gives us the property that linear combinations of physical amplitudes lead to new physically allowed amplitudes. To deal with the probability interpretation, we briey consider the denition of probability: Def: Probability: If S is a (sample) space, and P (E ) is a real additive set function dened on sets E in S , then P is referred to as a probability function if: 1. 2. 3. 4. If E is a subset (event) in S , then P (E ) 0. P (S ) = 1. If E, F S , and E F = , then P (E F ) = P (E ) + P (F ). If S is an innite sample space, we require that: P (E1 E2 . . .) = P (E1 ) + P (E2 ) + . . . for any sequence of disjoint events E1 , E2 , . . . in S . [For those with the mathematical background, we remark that a shorter denition for probability is: A probability function is a measure on S such that P (S ) = 1. We note that P is dened on all subsets E of S , hence is dened on a -ring.] Thus, the requirement of a probability interpretation means that any allowable wave function (s) dened on sample space S must be normalizable and square-integrable such that: (s) (s)(ds) = 1.
S
(28)
(29)
The integral here is in the Lebesgue-summable sense, and is the appropriate measure function on subsets of S . A measure function is simply a prescription for measuring the sizes of sets, implemented in a mathematically rigorous manner. 7
The mathematical considerations here are both critical to the foundation of quantum mechanics and potentially unfamiliar to the reader, so we will digress briey in order to develop an intuitive understanding of the need for them. Apparently, it is important to know how to measure the sizes of sets in our probability sample space. This is implemented abstractly in measure theory. We will not develop this here; a couple of examples should provide the intuition that is sucient for present purposes. For a rst example, suppose our sample space is the set of real numbers, R1 . In this case, the appropriate way to measure sizes of sets is a suitable generalization of our ordinary notion that the size of the interval (a, b) is just b a. This generalization is called the Lebesgue measure on R1 . It has the property that a denumerable set of discrete points is measureable, with measure zero. We remark that the Reimann integral is not suciently general for our purposes. A function f (x) is Riemann-integrable on [a, b] if and only if: f (x) is bounded. The set of points of discontinuity of f has Lebesgue measure zero. For example, the integral
0 1
f (x) dx,
(30)
where f (x) =
0 if x is rational, 1 if x is irrational,
(31)
is not dened. The function is discontinuous at every point, hence the measure of the points of discontinuity is non-zero: ({points of discontinuity}) = ({(0, 1)}) = 1. (32)
This is perhaps a pathological example. A more obvious example is that the Riemann sum doesnt allow us to sum over state variables with possibly discrete spectra, e.g., quantized energy levels. We could handle such situations in an ad hoc manner, but to build a rigorous foundation we resort to the Lebesgue integral. The idea of the Lebesgue integral is simple and elegant. Rather than divide the x-axis up into intervals, as in the Riemann integral, we divide the y -axis. That is, we partition the y -axis into intervals yi , i = 1, 2, . . . 8
Choose a point yi in each interval. Consider the sets f 1 (yi ). Multiply the measure of each such set by the corresponding yi. Then sum the products. The Lebesgue integral is the value of this sum in the limit where all of the yi intervals vanish.
f(x) f(x) y 6 y 5 y y 4 3y 3 y 2 y 1
{
{ {
a
x
1
x
2
... ...
b
x
a
f -1( y3) (b)
b
x
(a)
Figure 1: (a) The Riemann integral is a limit of slices in x. (b) The Lebesgue integral is a limit of slices in y .
IL lim
yi 0
yi [f 1 (yi )].
i
(33)
For this to work, f 1 (yi ) must be measurable sets, that is, f (x) must be a measurable functon: Def: A real function f (x), dened on S is said to be measurable if, for every real number u, the set Su = {x : f (x) < u, x S } is measurable. For example, consider the function of Eqn. 31. Take, in the limit, y1 = 0 and y2 = 1. Then f 1 (y1 ) = rational numbers f 1 (y2 ) = irrational numbers, 9 (34)
f(x)
u
Su
Figure 2: The set Su .
and 1 = ([0, 1]) = ({rationals on [0, 1]}) + ({irrationals on [0, 1]}) = 0 + ({irrationals on [0, 1]}). (35) Hence,
0 1
f (x) (dx) = 1
(36)
is the Lebesgue integral. The choice of measure (of the size of a set) may depend on the physical circumstance. We have used the Lebesgue measure on R1 , appropriate to continuous state variables. Another important measure is the Dirac measure (on S = R1 ): Let x0 R1 . The Dirac measure associated with point x0 is dened by: 1 if x0 E (37) (E ) = / 0 if x0 E .
10
Note that this is the appropriate measure to use for discrete state variables:
S
f (x) (dx) = f (x0 ).
(38)
We are still trying to build a suitable function space for our quantum mechanical wave functions. What about pathological functions, e.g., with many discontinuities. We will build into our space the concept that two functions that dier only in ways which will not aect observable probabilities are not to be considered as distinct. We proceed as follows: Def: A property, Q(x), which depends on location x in space S , is said to hold almost everywhere if the set of points for which Q does not hold has measure zero. Def: Two functions f1 (x), f2 (x), dened on S (that is, assume nite values at every point of S ) are said to be equivalent if f1 (x) = f2 (x) almost everywhere: f1 f2 . If equivalent functions f1 (x) and f2 (x) are integrable in the Lebesgue sense (summable) on a set E , then
E
f1 (x)(dx) =
E
f2 (x)(dx).
(39)
Thus, if we decompose the set of summable functions into classes of equivalent functions, the integral can be regarded as a functional dened on the space F , of these classes.
3.1
The Space L2
For our quantum mechanical wave functions we are of course interested in complex functions. A complex function f (x) = f1 (x) + if2 (x), where f1 and f2 are real functions, is said to be summable on E if f1 and f2 are summable: f (x)(dx) =
E E
f1 (x)(dx) + i
E
f2 (x)(dx).
(40)
Theorem: A complex function f (x) is summable if and only if its absolute value, |f (x)| = f1 (x)2 + f2 (x)2 , (41) is summable. 11
Proof: Suppose f (x) = f1 (x) + if2 (x) is summable on E. Then f1 and f2 are summable on E . By virtue of our dention of the integral, Eqn. 33, |f1 | and |f2 | are therefore also summable. Hence,
E
|f (x)|(dx) =
E E
|f1 (x) + if2 (x)|(dx) |f1 (x)| + |f2 (x)|(dx), by the triangle inequality (42)
< .
Conversely, suppose |f | is summable on E . Then, again referring to the denition in Eqn. 33, f (x)(dx)
E
< .
E
|f (x)|(dx) (43)
Even for complex functions, the integral denes a linear functional. Let L denote the space of complex functions f (x) such that |f (x)|2 is summable on S : |f (x)|2(dx) < , for f (x) L. (44)
S
Theorem: The space L is a linear space (or vector space). Proof: The principal step of the proof is as follows: Suppose f (x) L and g (x) L. Then |f (x) + g (x)|2 = 2|f (x)|2 + 2|g (x)|2 |f (x) g (x)|2 2|f (x)|2 + 2|g (x)|2.
(45)
But 2|f (x)|2 + 2|g (x)|2 is summable, and hence |f (x) + g (x)|2 is summable. The space L is our candidate space for physical quantum mechanical wave functions. However, there is a problem with it: There are distinct elements of L, diering on sets of measure zero, which correspond to the same physics. Let us tidy this ugliness up. Consider Z the subset of L consisting of functions f (x) such that
S
|f (x)|2 (dx) = 0. 12
(46)
Note that Z is a linear subspace of L, since if f Z , then kf Z for all complex constants k , and if f, g Z , then f + g Z since |f + g |2 2|f |2 + 2|g |2. Thus, we can dene the factor space, L2 = L/Z. (47)
That is, two functions fa (x), fb (x) in L determine the same class in L2 if and only if the dierence fa fb vanishes almost everywhere, i.e.,
S
|fa (x) fb (x)|2 (dx) = 0.
(48)
We say that the space L2 consists of functions f (x) such that |f (x)|2 is summable on S , with the understanding that equivalent functions are not considered distinct. In other words, L2 is a space of equivalence classes. Finally, we add to this space the notion of a scalar product. We start by noting that the product of two elements of L2 is summable: Theorem: If f, g L2 , then f g is summable on S . Proof: Write f g = 1 |f + g |2 |f g |2 + i|f ig |2 i|f + ig |2 4 (49)
Each term on the right is summable, and hence the product f g L2 . Theorem: L2 is a Hilbert space, with scalar product dened by: f |g f (x) g (x)(dx).
S
(50)
Proof: The proof starts by showing that L2 is a pre-Hilbert space, that is, a linear space upon which a scalar product has been properly dened. This consists in showing that: 1. f |f = 0 if and only if f = 0. The fact that L2 is a space of equivalence classes is crucial here. 2. f |g = g |f . 3. f |cg = c f |g . 4. f |g1 + g2 = f |g1 + f |g2 . 13
Once it has been demonstrated that L2 is a pre-Hilbert space, it remains to show that L2 is complete. That is, it must be shown that every Cauchy sequence of vectors in L2 converges to a vector in L2 . A fundamental postulate of quantum mechanics is: To every physical system S there corresponds a separable Hilbert space, HS . L2 appears to be a suitable space for our probability amplitudes since it is a linear space (hence we have superposition), and its elements are normalizable (square-summable, hence can make a probability interpretation). The addition of the scalar product permits us to make projections in our vector space. Note tht L itself was not sucient for this construction, since f |f = 0 is not equivalent to f = 0 in L. It should be understood that using L2 is all right, since functions which dier only on sets of measure zero will yield the same probabilistic, and hence physical, results. The availability of the scalar product in particular leads to the possibility of constructing a (orthonormal) basis. Completeness means that we have included a suciently large set of vectors that we wont encounter diculties when we consider certain sequences of vectors. We can construct a complete orthonormal basis {|e } on a Hilbert space such that every vector |x H can be expanded: |x = |e e |x . (51)
Abstractly, a separable space is a topological space T which contains a denumerable (countable) set of points {t1 , t2 , . . .} which is dense in T . The point of the postulate that the Hilbert space corresponding to any physical system be separable is that there is then a denumerable dense set of vectors. We may nd a complete denumerable basis in which to expand our vectors. To complete the connection of this postulate with our space L2 we have the theorem: 14
Theorem: The space L2 (a, b) (where it is permissible for a = , b = ) with Lebesgue measure is separable. Proof: To prove this theorem, we rst prove that there exists a complete denumerable orthonormal basis in L2 . For example, on L2 (0, 2 ), the set of functions: eikx , k = 0, 1, 2, . . . , (52) 2 forms a complete orthonormal system. Then we show that from this basis we can construct a countable dense set of vectors in L2 . It may be noted that non-separable Hilbert spaces do exist. However, we have so far not found a need to consider them for quantum mechanics. 2 On L2 (, ), with measure (dx) = ex dx, the Hermite polynomials: Hn (x) = d n x 2 e, n 2n n! dx ex
2
n = 0, 1, . . .
(53)
form a complete orthonormal system. Alternatively, with measure (dx) = 2 dx, the functions ex /2Hn (x) form a complete orthonormal system.
4
Observables
An observable Q is a physical quantity. In quantum mechanics, we deal with the probability p(Q, ) that a measurement will yield a value of Q in a subset of the set of real numbers. A fundamental postulate of quantum mechanics is that: Every observable corresponds to a self-adjoint operator dened in HS . The term dened in HS means, for operator Q, that x DQ HS , and Qx RQ HS , where DQ is the domain of the operator, and RQ is its range. Self-adjoint operators are evidently an important class of operator the key point is that a self-adjoint operator is also a Hermitian operator, and hence has a real eigenvalue spectrum. This is the physical reason why they are of interest. Let us look at some of the mathematical aspects.
15
Def: (Adjoint) Let L be a linear operator, dened in HS with domain DL , such that DL is dense in HS (that is, DL = HS , where DL is the closure 1 of DL ). The adjoint, L , of L is dened by: L u|v = u|Lv , v DL . (54)
In other words, u is a vector in HS such that there exists a w HS satisfying u|Lv = w |v . If this holds, then we say w = L+ u; the adjoint operator maps u to w . The requirement that DL be dense in is HS necessary in order for L to be uniquely dened. To see this, suppose that it is not unique, i.e., suppose there exist two vectors wa , wb such that u|Lv = wa |v = wb |v , v DL . (55)
In this case, (wa wb )|v = 0. But wa wb is thus orthogonal to every vector in a dense set, and therefore wa wb = 0. This last point could use some further proof; well depend on its evident plausibility here. Def: Self-adjoint: If L = L (which means: DL = DL , and L u = Lu for all u DL ), then L is said to be self-adjoint. Note the distinction between a self-adjoint operator, and a Hermitian operator, dened according to: Def: Hermitian: A linear operator L, with DL HS , is called Hermitian if Lu|v = u|Lv , u, v DL . (56) For example, in the case of a nite dimensional vector space, L is a square matrix, and we have: u|Lv Lu|v = = = = u Lv (Lu) v u L v u Lv if L = L. (57) (58) (59)
There are a variety of notations used to denote the adjoint of an operator, most notably L , L+ , and L . Well adopt the dagger notation here, as it is consistent with the familiar complex-conjugatetranspose notation for matrices. The asterisk notation is common also, but we avoid it here on the grounds of potential confusion with simple complex conjugation.
1
16
In this case, a self-adjoint operator is also a Hermitian operator: L u|v = Lu|v = u|Lv , u, v DL = DL . (60)
However, a Hermitian operator is not necessarily a self-adjoint operator, if the space is innite dimensional. The issue is one of domain. It can happen, in an innite dimensional Hilbert space, that a Hermitian operator, L, has DL DL+ , as a proper subset. In this case, L is not self-adjoint. Consider an example to illustrate this inequivalence. A dierential equation (where L is a dierential operator, and we write Lu = a) is not completely specied until we give certain boundary conditions which the solution must satisfy. Thus, for a function u to belong to DL , not only must the expression Lu be dened, but u must also satisfy the boundary conditions. If the boundary conditions are too restrictive, we might have DL DL+ but DL = DL+ , so that a Hermitian operator may not be self-adjoint. To illustrate with a specic example, let L = p be the momentum operator in one dimension: 1d p= , x [a, b]. (61) i dx The boundary conditions are to be specied, but the domain of this operator is otherwise the set of continuous functions on [a, b]. This set of functions is dense in our Hilbert space L2 (a, b). We look at the scalar product of pv with u, where u and v are continuous functions: u|pv = =
b a
u (x)
1d v (x) dx i dx
(62)
b 1d 1 u (x)v (x)|b u (x) v (x) dx a i i dx a 1 [u (b)v (b) u (a)v (a)] + pu|v = i = p u | v
(63) (64)
The u (b)v (b) u (a)v (a) boundary term portion of Eqn. 63 must vanish for all v Dp in order for p to be a Hermitian operator; hence we shall assume this condition. There is, however, more than one way to achieve this, even with the dense requirement. For example, we could impose the boundary condition v (a) = v (b) = 0, so that Dp = {v |v is continuous, and v (a) = v (b) = 0}. In this case, u need not satisfy any constraints at a or b, and 17
Dp+ = {u|u is continuous} = Dp , since p+ u|v = u|pv = 1d u |v , i dx (65)
d for all continuous functions u. We have p = 1 dx with Dp = {u|u continui ous on [a, b]}. So, p is Hermitian, but not self-adjoint, since Dp is a proper subset of Dp . On the other hand, if we had chosen the extension of the above p with boundary condition v (a) = v (b), then we would nd a restriction of the above p , with Dp = {u|u(a) = u(b), u continuous on [a, b]}. With this denition p is a self-adjoint operator.
5
The Uncertainty Principle
The famous uncertainty principle is discussed in every introductory quantum mechanics course. We revisit it briey here. First, the reader is reminded of the important Schwarz inequality: Theorem: For any vectors , in our Hilbert space, | | | | | . (66)
Equality holds if and only if and are linearly dependent: = c , where c is a complex number. Proof: One way to prove the Schwarz inequality is to consider the nonnegative denite scalar product: + rei | + rei 0. (67)
Expanding the left hand side results a quadratic expression for r . Considering the possible solutions for r yields a constraint on the discriminant. The resulting inequality is the Schwarz inequality. Suppose now that we have two self-adjoint operators A and B , and a state vector in the domains of both. The average (mean) value (expectation value) of observable A if the system is in state is: A = |A , 18 (68)
where we assume that is normalized. Likewise, the mean of observable B is B = |B . We are presently interested in learning something about the spreads of the distributions of observations of A and B . Thus, it is convenient to subtract out the means by dening shifted operators: AS A A BS B B . (69) (70)
The domains of the shifted operators are the same as the domains of the unshifted operators. We immediately have that AS = BS = 0. Dene the commutator of AS and BS : [AS , BS ] AS BS BS AS = [A, B ]. (71)
It should be noted that the product of two operators is dened by their operation on a state vector: AB | means rst apply operator B to , then apply A to the result. The obvious questions of domain need to be dealt with, of course. Thus, let us further require DAB , DBA , and consider: | [A, B ] | = (72) | |AS BS |BS AS | | |AS BS | + | |BS AS | (triangle inequality) | AS |BS | + | BS |AS | (self-adjointness) 2| AS |BS | AS |AS BS |BS
2 |A2 |BS S
2 2
(Schwarz inequality) (73)
(self-adjointness).
The variance of a distribution is a measure of its spread. For an observable Q, the variance for a system in state is dened by:
2 Q |(Q Q )2 = Q2 Q 2 .
(74)
The square root of the variance, Q is called the standard deviation. We 2 see that, for example, A = |A2 . Thus, we may rewrite Eqn. 73 in the S form: 1 A B | [A, B ] |. (75) 2 This is a precise statement of the celebrated uncertainty principle. We shall often use the convenient notation A A . The physical interpretation is 19
that, if we have two non-commuting observables, the product of the variances of the probability distributions for these two observables is bounded below. This is typically interpreted further with statements such as the ability to measure both variables simulataneously is limited. A measurement of one observable disturbs the system in a way that aects the result of a second measurement of the other observable. While there is some justication for such statements, one must be careful not to carry them too far in case of confusion, come back to what the principle actually says! For example, the ability to measure carries a connotation that there may be an issue of experimental resolution involved. While expermental resolution generally needs to be folded into the analysis of an actual experiment, it has nothing to do with the present point.
5.1
Example: Angular Momentum
The angular momentum operator for a particle is L = x p, where x is the position operator, and p is the momentum operator. This may be expressed in components as: Li = ijk xj pk . (76) The summation convention is used here: a sum is implied over repeated indices, in this case, there is a sum over j, k = 1, 2, 3. The quantity ijk is known as the antisymmetric symbol: i, j, k =cylic permutation of 1, 2, 3, = 1 i, j, k =anti-cylic permutation of 1, 2, 3, 0 any two indices the same.
+1
ijk
(77)
We remark that L = x p = p x are both acceptable, since only commuting components of x and p are paired. We know that [pm , xn ] = imn . (78) We are interested in the commutation relations of the angular momentum operators: [L , L ] = jk mn [xj pk , xm pn ] = i( jm jn jn jm )xm pn , (79) (80)
20
where the algebra between Eqns. 79 and 80 is left as an exercise for the reader. The reader is also encouraged to demonstrate that E,mn
jm jn
jn jm
=
j mjn .
(81)
With the identity of Eqn. 81, we obtain [L , L ] = i
L .
(82)
Thus, the uncertainty relation between components of angular momentum is: 1 1 (83) L L | [L , L ] | = | L |. 2 2 Let us illustrate this with an explicit example. We rst anticipate the generalization of angular momentum to include spin, with the same commutation relations, and consider the simplest system with non-zero angular momentum, spin-1/2. Well follow common convention, and pick our basis to be eignevectors of J3 (using now J to indicate angular momentum, leaving L to stand for orbital angular momentum). We again anticipate the quantization of spin, where the eigenvalues of J3 are 1/2 for a spin-1/2 system. In this basis, our angular momentum operator is: 1 J = , 2 where are the Pauli matrices: 01 , 2 = 1 = 10 0 i , i0 10 . 0 1 (84)
3 =
(85)
These are Hermitian matrices, hence correspond to observables. Suppose, in this basis, we have the state 1 = 2 1 , 1 (86)
which is a superposition of J3 = 1/2 eigenstates. We may compute expectation values of angular momentum for this state: J1 J2 J3 1 (1, 1) 4 1 (1, 1) = 4 1 (1, 1) = 4 = 0 1 0 i 1 0 21 1 1 1 = 0 1 2 i 1 =0 0 1 0 1 = 0. 1 1 (87) (88) (89)
2 To obtain the second moments, we notice that i = 1, i = 1, 2, 3. Thus,
1 Ji2 = , i = 1, 2, 3, 4 hence (J1 )2 = (J2 )2 = (J3 )2
2 J1 J1 2 2
(90)
=
1 1 4 2
= 0,
(91) (92) (93)
1 0 = 4 1 0 = = 4
1 , 4 1 . 4
Let us check the uncertainty relation involving J1 and J2 : J1 J2 = 0 1 1 = 0 | J3 | = 0 . 2 2 (94)
So this relation is satised. Physically, it may readily be seen that our state is actually an eigenstate of J1 with eigenvalue 1/2. It is a superposition of J2 = 1 eigenstates. Even though our lower bound on the product of uncertainties 2 is zero, and is achieved, we cannot measure J1 and J2 simultaneously with arbitrary precision. As soon as we know J1 = 1/2, a measurement of J2 will yield 1/2 with equal probability. Alternatively, if we rst measure J2 , obtaining a value of either 1/2 or 1/2, a subsequent measurement of J1 will yield 1/2 with equal probability. The measurement of J2 has disturbed the state. It should perhaps be remarked that the term precision here is in the frequency sense: Imagine that you can prepare the identical state many times and repeat the measurements. The measurements will yield dierent results among the samplings, with expectation values as we have calculated, in the limit of averaging over an innite number of samplings. Finally, let us also look at: J2 J3 = 11 1 1 1 = | J1 | = . 22 4 2 4 (95)
Again, the uncertainty principle is satised.
22
6
Exercises
1. Show that L2 is complete. 2. Complete the proof that the space L2 (a, b) is separable. 3. Show that if x H , where H is a separable Hilbert space, is orthogonal to every vector in a dense set, then x = 0. 4. Complete the proof of the Schwarz inequality. 5. Complete the derivation of Eqns. 80, 81, and 82. 6. Time Reversal in Quantum Mechanics: We wish to dene an operation of time reversal, denoted by T , in quantum mechanics. We demand that T be a physically acceptable transformation, i.e., that transformed states are also elements of the Hilbert space of acceptable wave functions, and that it be consistent with the commutation relations between observables. We also demand that T have the appropriate classical correspondence with the classical time reversal operation. Consider a system of structureless (fundamental) particles and let X = (X1 , X2 , X3 ) and P = (P1 , P2 , P3 ) be the position and momentum operators (observables) corresponding to one of the particles in the system. The commutation relations are, of course: [Pm , Xn ] = imn , [Pm , Pn ] = 0, [Xm , Xn ] = 0. The time reversal operation T : t t = t, operating on a state vector gives (in Schrdinger picture you may consider how to make o the equivalent statement in the Heisenberg picture): T | (t) = | (t ) . The time reversal of any operator, Q, representing an observable is then: 23
Q = T QT 1 (a) By considering the commutation relations above, and the obvious classical correspondence for these operators, show that T iT 1 = i. Thus, we conclude that T must contain the complex conjugation operator K : KzK 1 = z , for any complex number z , we require that T on any state yields another state in the Hilbert space. We can argue that (for you to think about) we can write: T = UK , where U is a unitary transformation. If we operate twice on a state with T , then we should restore the original state, up to a phase: T 2 = 1, where is a pure phase factor (modulus = 1). (b) Prove that = 1. Hence, T 2 = 1. Which phase applies in any given physical situation depends on the nature of U , and will turn out to have something to do with spin, as we shall examine in the future. 7. Let us consider the action of Gallilean transformations on a quantum mechanical wave function. We restrict ourselves here to the proper Gallilean Transformations: (i ) translations; (ii ) velocity boosts; (iii ) rotations. We shall consider a transformation to be acting on the state (not on the observer). Thus, a translation by x0 on a state localized at x1 produces a new state, localized at x1 + x0 . In conguration space, x x we have a wave function of the form (x, t). A translation T (x0 ) by x0 of this state yields a new state (please dont confuse this translation operator with the time reversal operator of the previous problem, also denoted by T , but without an argument): x x x (x) = T (x0 ) (x, t) = (x x0 , t). 24 (96)
Note that we might have attempted a denition of this transformation with an additional introduction of some overall phase factor. However, it is our interest to dene such operators as simply as possible, consistent with what should give a valid classical correspondence. Whether we have succeeded in preserving the appropriate classical limit must be checked, of course. Consider a free particle of mass m. The momentum space wave function is itp2 p p (p, t) = f (p) exp , (97) 2m where p = |p|. The conguration space wave function is related by the (inverse) Fourier transform: x (x, t) = 1 (2 )3/2
()
p p d3 (p)eixp (p, t).
(98)
Obtain simple transformation laws, on both the momentum and conguration space wave functions, for each of the following proper Gallilean transformations: x (a) Translation by x0 : T (x0 ) (note that we have already seen the result in conguration space). (b) Translation by time t0 : M (t0 ). v (c) Velocity boost by v0 : V (v0 ). (Hint: rst nd v p (p, 0) = f (p) = V (v0 )f (p), then etc.) (d) Rotation about the origin given by 3 3 matrix R: U (R). Make sure your answers make sense to you in terms of classical correspondence. 8. Consider the (real) vector space of real continuous functions with continuous rst derivatives in the closed interval [0, 1]. Which of the following denes a scalar product? 25
2 (p, t) = f (p)eitp /2m ,
(99) (100)
(a) f |g = (b) f |g =
1 0 1 0
f (x)g (x)dx + f (0)g (0) f (x)g (x)dx
9. Consider the following equation in E (innite-dimensional Euclidean space let the scalar product be x|y x yn ): n=1 n Cx = a, where the operator C is dened by (in some basis): C (x1 , x2 , . . .) = (0, x1 , x2 , . . .) Is C : (a) A bounded operator [i.e., does there exist a non-negative real number such that, for every x E , we have |Cx| |x| (|x| denotes the norm: x|x )]? (b) A linear operator? (c) A hermitian operator (i.e., does x|Cy = C x|y )? (d) Does Cx = 0 have a non-trivial solution? Does Cx = a always have a solution? Now answer the same questions for the operator dened by: G(1 , 2 , . . .) = (1 , 2 /2, 3/3, . . .). (101)
Note that we require a vector to be normalizable if it is to belong to E i.e., the scalar product of a vector with itself must exist. 10. Let f L2 (, ) be a summable complex function on the real interval [, ] (with Lebesgue measure). (a) Dene the scalar product by: f |g =
f (x)g (x)dx,
(102)
for f, g L2 (, ). Starting with the intuitive, but non-trivial, assumption that there is no vector in L2 (, ) other than the 26
trivial vector (f 0) which is orthogonal to all of the functions sin(nx), cos(nx), n = 0, 1, 2, . . ., show that any vector f may be expanded as:
f (x) =
n=0
(an cos nx + bn sin nx) ,
(103)
where a0 = an bn 1 f (x)dx 2 1 = f (x) cos nxdx 1 = f (x) sin nxdx. (104) (n > 0) (105) (106)
[You may consult a text such as Fanos Mathematical Methods of Quantum Mechanics for a full proof of the completeness of such functions.] (b) Consider the function: x < 0, f (x) = 0 x = 0, +1 x > 0.
1
(107)
Determine the coecients an , bn , n = 0, 1, 2, . . . for this function for the expansion of part (a). (c) We wish to investigate the partial sums in this expansion:
N
fN (x) =
n=0
(an cos nx + bn sin nx) .
(108)
Find the position, xN of the rst maximum of fN (for x > 0). Evaluate the limit of fN (xN ) as N . Give a numerical answer. In so doing, you are nding the maximum value of the series expansion in the limit of an innite number of terms. [You may nd the following identity useful:
N n=1
cos(2n 1)x =
1 sin 2Nx .] 2 sin x
(109)
27
(d) Obviously, the maximum value of f (x), dened in part (b), is 1. If the value you found for the series expansion is dierent from 1, comment on the possible reconciliation of this dierence with the theorem you demonstrated in part (a). 11. Show that, with a suitable measure, any summation over discrete indices may be written as a Lebesgue integral:
n=1
f (xn ) =
{x}
f (x)(dx).
(110)
12. Resonances II: Quantum mechanical resonances Earlier we investigated some features of a classical oscillator with a resonant behavior under a driving force. Let us begin now to develop a quantum mechanical analogue, of relevance also to scattering and particle decays. For concreteness, consider an atom with two energy levels, E0 < E1 , where the transition E0 E1 may be eected by photon absorption, and the decay E1 E0 via photon emission. Because the level E1 has a nite lifetime we denote the mean lifetime of the E1 state by it does not have a precisely dened energy. In other words, it has a nite width, which (assuming that E0 is the ground state) can be measured by measuring precisely the distribution of photon energies in the E1 E0 decay. Call the mean of this distribution 0 . (a) Assume that the amplitude for the atom to be in state E1 is given by the damped oscillatory form: (t) = 0 ei0 t 2 Show that the mean lifetime is given by , as desired. (b) Note that our amplitude above satises a Schrdinger equation: o i i d (t) = (0 ) (t) dt 2
t
Suppose we add a sinusoidal driving force F eit on the right hand side, to describe the situation where we illuminate the atom with monochromatic light of frequency . Solve the resulting inhomogeneous equation for its steady state solution. 28
(c) Convince yourself (e.g., by conservation of probability) that the intensity of the radiation emitted by the atom in this steady-state situation is just | (t) |2 . Thus, the incident radiation is scattered by our atom, with the amount of scattering proportional to the emitted radiation intensity in the steady state. Give an expression for the amount of radiation scattered (per unit time, per unit amplitude of the incident radiation), as a function of . For convenience, normalize your expression to the amount of scattering at = 0 . Determine the full-width at half maximum (FWHM) of this function of , and relate to the lifetime . Note that the Breit-Wigner function is just the Cauchy distribution in probability. 13. Time Reversal in Quantum Mechanics, Part II We earlier showed that the time reversal operator, T , could be written in the form: T = UK, where K is the complex conjugation operator and U is a unitary operator. We also found that T 2 = 1. Consider a spinless, structureless particle. All kinematic operators for such a particle may be written in terms of the X and P operators, where [Pj , Xk ] = ijk T XT 1 = X T P T 1 = P (where the latter two equations follow simply from classical correspondence). If we work in a basis consisting of the eigenvectors of X , the eigenvalues are simply the real position vectors, and hence: U XU 1 = X. In this basis, the matrix elements of P may be evaluated: P = i : 29
x1 | P | x2
=
()
(3) (x x1 )(ix ) (3) (x x2 )d(3) x
= ix1 (3) (x1 x2 ). Thus, these matrix elements are pure imaginary, and K P K 1 = P , which implies nally U P U 1 = P .
We conclude that for our spinless, structureless particle: U = 1ei , where the phase may be chosen to be zero if we wish. In any event, we have: T = ei K, and T 2 = ei Kei K = 1.
(a) Show that, for a spin 1/2 particle, we may in the Pauli representation (that is, an angular momentum basis for our spin-1/2 system such that the angular momentum operators are given by one-half the Pauli matrices) write: T = 2 K, and hence show that: T 2 = 1.
Note that the point here is to consider the classical correspondence for the action of time reversal on angular momentum. By considering a direct product space made up of many spin-0 and spin 1/2 states (or by other equivalent arguments), this result may be generalized: If the total spin is 1/2-integral, then T 2 = 1; otherwise T 2 = +1.
30
(b) Show the following useful general property of an antiunitary operator such as T : Let | = T | | = T | . Then | = | . This, of course, should agree nicely with your intuition about what time reversal should do to this kind of scalar product. (c) Show that, if | is a state vector in an odd system (T 2 = 1), then T | is orthogonal to | . 14. Suppose we have a particle of mass m in a one-dimensional potential V = 1 kx2 (and the motion is in one dimension). What is the minimum 2 energy that this system can have, consistent with the uncertainty principle? [The uncertainty relation is a handy tool for making estimates of such things as ground state energies.]
31