Principles of Quantum Mechanics

Home , Position operator, Probability current

MATH3385/5385. Quantum Mechanics. Handout # 3: The Quantum Mechanics of Structureless Particles

Here we will study the principles of the QM of a structureless particle subject to a (possibly time-dependent) potential V (r,t). Structureless means that there are no other character- istics of the particle other than its position and momentum ( in contrast to e.g. particles with spin, a characteristic that we will introduce at a later stage). Our considerations con- cerning the wave packet description (see Handout # 2) has demonstrated that contrary to the classical perception of a particle (i.e. a point-like object which follows a geometric trajectory characterised at each point in time by its position and velocity) a fundamental uncertainty must be part of the new theory: in QM no longer can we measure both position as well as momentum of a particle simultaneously to arbitrary precision. This imprecision is not a fault of our apparatus or measuring device: it is fundamental to the theory. If we endeavour to measure the position, we lose track of the momentum and vice versa. We have to come to terms with this new and unfamiliar point of view on particles in the quantum theory. To capture both the particle aspect as well as the wave aspect, we need a description that achieves two things:

• it allows for superposition of waves, thus incorporating the wave aspect, and

• it has a built-in conservation law which accounts for the particle aspect (indeed: the particle is preserved; it is not supposed to vanish).

We will now highlight the various ingredients of the theory.

Wave function and probability interpretation

The main idea behind quantum mechanics is to describe particles by a complex valued wave function ψ(r,t) which represents the spatial state of the particle. The word state is one of the key words in QM: instead of a particle having at each time t a precise value associated with it for the physical observables such as position and momentum, we say that the particle is prepared in a certain state described by the wave function at time t. What does the wave function represent physically? The interpretation (according to the Copenhagen school1) is the following: the absolute value P (r,t)= |ψ(r,t)|2 is a probability density, meaning that

1This interpretation, which was promoted by N. Bohr and M. Born, has not remained undisputed. In fact, A. Einstein objected very strongly against the probability interpretation, saying that “God does not play dice”. Even today the discussions surrounding the probability interpretation of QM is carrying on. However, irrespective of the interpretation, the mathematical consistency of the theory is well-established.

1 |ψ(r,t)|2dr = the probability of ﬁnding a particle at time t in an inﬁnitesimally small box of volume dr = dx dy dz located at r.

This is quite a new and revolutionary idea: it introduces an amount of subjectiveness in the description of nature. The “observer” (i.e. the person or the device that performs a measurement), is no longer isolated from the object it observes (i.e. the particle), but inﬂuences in a fundamental way the state in which the particle is found. In fact, the act of performing a measurement actually prepares the particle in a given state, namely in the state where the measured quantity is relatively well-localised (the bulk of the wave packet lying within the error measurement from the value predicted). Thus, a measurement invokes an irreversible change in the state of the particle. However, moments after the measurement, the wave packet evolves and starts to spread as we lose track of the particle, and we need to perform another measurement to localise the particle once again (in either the momentum or position space). This procedure can be graphically represented as follows

↑ ↑ −→ measurement measurement time

The quantum mechanical evolution of a particle: the dashed cloud represents the wave distribution of the particle in space, the fuzziness of which indicates the imprecision before measurement. A measurement localises the wave function, after which the wave packet spreads out, and another measurement is needed to pinch the wave packet again, etc.

By the deﬁnition of probability, the total probability (i.e. the sum over all values of the relevant, i.e. measured, variables characterising the particle) must be equal to one. In the present case where the wave function is a function of the position we have:

∞ ∞ ∞ P (r,t) dr = 1 , dr = dx dy dz Z Z Z−∞ Z−∞ Z−∞ where the integral represents the integration over three-dimensional space. This expresses the fact that requires that the wave functions are square-integrable i.e. the integral

|ψ(r,t)|2 dr < ∞ Z and we will work with wave functions that are normalised to one2. The matter in which function spaces ψ takes values is a rather subtle one and we leave this to later.

2 2 If the wave function is not normalised and we have R |ψ(r,t)| dr = C we can always redeﬁne ψ by dividing by C1/2 such that it becomes normalised as long as ψ is square integrable.

2 The complexity of the wave function ψ allows us to explain the nontrivial interference patterns that were discussed in Handout # 2. Obviously, ψ itself is not a measurable quantity because in physics we can only measure real-valued objects. Thus, it seems that the phase factor exp(iθ) associated with the wave function

ψ(r,t)= |ψ(r,t)|eiθ(r,t) is not of physical importance. This, however is an erroneous point of view because when iθ1 iθ2 one superimposes two waves ψ1 = |ψ1|e and ψ2 = |ψ2|e , the superimposed wave function does depend on the relative phase θ1 − θ2 of the waves. In fact, we have

2 2 iθ1 iθ2 |ψ1 + ψ2| = |ψ1|e + |ψ2|e

2 2 ∗ = |ψ1| + |ψ2| + 2Re( ψ1ψ2) 2 2 i(θ1−θ2) = |ψ1| + |ψ2| + 2|ψ1| |ψ2| Re e 2 2 = |ψ1| + |ψ2| + 2|ψ1| |ψ2| cos(θ1 − θ2)

Note that the complexity of the wave function is essential in the appearance of the relative phase factor: we do not see it in the individual waves since |eiθ| = 1, but it appears in their interference.

Evolution of the system and conservation law

The main equation that governs the time evolution of the wave function is the Schr¨odinger equation (SE). We have already derived a simple case of the SE in Handout # 2 in the case of a one-dimenional free particle. In the general case of a particle moving in three- dimensional space subject to a potential V (r,t) (possibly time-dependent) we have the following equation:

The Schr¨odinger Equation:

~ ∂ψ ~2 ∇2 ∇2 ∂2 ∂2 ∂2 i ∂t = − 2m ψ + V (r,t)ψ , = ∂x2 + ∂y2 + ∂z2

One important aspect of the SE is that it preserves probability. This is important, because the total probability should be a constant of the motion: once we know that the particle exists it will not disappear and thus we have certainty (i.e. probability equal to one) of ﬁnding the particle somewhere. This can easily seen as folows. Using the SE, and its complex conjugate, i.e.

∂ψ∗ ~2 −i~ = − ∇2ψ∗ + V (r,t)ψ∗ ∂t 2m we derive by multiplying the SE by ψ∗ and the above equation by ψ and then subtract:

∂ψ ∂ψ∗ ~2 i~ ψ∗ + ψ = − ψ∗∇2ψ − (∇2ψ∗)ψ ∂t ∂t 2m

3 where the terms containing the potential V cancel eachother. This can be rewritten as:

∂ ~2 i~ |ψ|2 = − ∇ · (ψ∗∇ψ − (∇ψ∗)ψ) ∂t 2m which leads to the conservation law: ∂ i~ P + ∇ · j = 0 , j = − (ψ∗∇ψ − (∇ψ∗)ψ) ∂t 2m and where as before P (r,t)= |ψ(r,t|2. The vector quantity j is called the probability current, whereas P is the probability density. The conservation law leads to the conservation of probability as follows: let us integrate the formula over a volume V (after which we let V grow inﬁntely large so that we get an integral over the entire three-dimensional space). This leads to d P (r,t) dr = − ∇ · j dr = − j · dS dt ZV ZV ZS where we have used the divergence theorem (see MATH2360: vector calculus) to obtain an integral over the closed surface S that surrounds the volume V. Thus this integral depends only on the values of j (and hence of ψ and ψ∗) on the surface S. Since ψ is assumed to be quadratically integrable, and consequently ψ → 0 as |r|→∞ the integral on the right-hand side vanishes as we let the volume V grow. Thus, we obtain for the integrals over the entire space: d P (r,t) dr = 0 ⇒ P (r,t) dr = constant dt Z Z and consequently the total probability is preserved. This tells us that the Schr¨odinger equation is consistent with the normalisation of the total probability to one.

Physical observables and expectation values

In QM as in classical mechanics we want to be able to calculate the value of physical observables. In classical mechanics this amounts to trying to predict the values of all relevant variables (position, momentum, etc.) by solving the equations of the motion, and nothing prevents us to do that in principle to arbitrary precision. In QM we can no longer do that: not all relevant variables can be simultaneously measured nor predicted to arbitrary precision: some physical quantities are incommensurable. If two observables (i.e. physical quantites that we can actually measure) are in principle simultaneously measurable then we call them commensurable and we can fully characterize the state of the particle by giving values to a large enough set of commensurable quantities. However, some quantities (like the corresponding components of momentum and position) are incommensurable and there will always be a fundamental imprecision in the measurement of one or the other, given by the uncertainty relations, e.g. ∆x∆px & ~/2. So, the actual value of an observable has a diﬀerent meaning in QM than in classical mechanics. In view of the probability interpretation of quantum mechanical states, what we can always do is calculate an average value in the sense of probability theory of a physical quantity. This is called the expectation value of an observable, and it corresponds to the

4 most expected outcome of a measurement of that quantity in experiment. The mathematical deﬁnition of an expectation value, e.g. of a function f(r) of position, is inspired by probability theory (see Appendix B) and is given by the following:

hf(r)i = f(r)P (r,t) dr Z Note that the expectation value hfi only depends on t, but not on r since we perform an integration over r. If we would, many times over, do an experiment measuring the observable f then we would ﬁnd various outcomes, but the average value of these experiments would be predicted to coincide with this expectation value. Nonetheless, in realistic experiments there would be a spreading of values, so we need to have measure for the spreading as well. This is again inspired by probability theory where one would use the root mean-square deviation to measure the width of the spreading of experiments:

∆f = h( f − hfi )2i p which is the square root of the so-called variance in probability theory. Inspired by probabilty theory we can also introduce the covariance between observables f(r) and g(r) which is given by the expectation value:

Cov(f, g)= h (f − hfi)(g − hgi) i = hfgi − hfihgi which is a measure for what in physics is called the correlation between the physical quantities f and g. If we are interested in calculating expectation values for quantities depending on momentum p rather than position we need to work in another representation of the state of the particle: the momentum representation. More generelly, if one is interested in the expectation value of mixed physical quantities depending on both position as well as momentum (such as for instance the angular momentum of the particle, i.e. ℓ = r × p , see Handout # 1) we need to represent such quantities in any representation. This requires the introduction of physical observables as operators acting on some conveniently chosen Hilbert space of wave functions (see Appendix C).

Representations: Momentum and position representation

As said earlier, position and momentum of a particle are incommensurable observables: we cannot measure them simultaneously to arbitrary precision. In fact, the state of a particle can be represented in the position representation by the wave function ψ(r,t) as before, but we can equally well decsribe the particle in terms of a momentum wave function ϕ(p,t) depending on the momentum p. Similarly as for the position representation, the thus obtained momentum representation of the particle has a probability interpretation: the absolute value |ϕ(p,t)|2 is the momentum probability density, meaning that

|ϕ(p,t)|2dp = the probability of ﬁnding a particle at time t in an inﬁnitesimally small box of volume dp = dpx dpy dpz in momentum space located at p.

5 Thus, again since the probabilty of the particle having any momentum is equal to one we need to require that the momentum wave functions are also square-integrable

∞ ∞ ∞ Q(p,t) dp < ∞ , dp = dpx dpy dpz Z Z Z−∞ Z−∞ Z−∞ where Q(p,t) = |ϕ(p,t)|2 is the momentum probability density, which allows us to nor- malise the wave function ϕ to unity. Working in the momentum representation goes in precisely the same way as before: for normalised wave functions ϕ(p,t) we can calculate expectation values of momentum dependent quantities by the formula:

hg(p)i = g(p)Q(p,t) dp Z and the spreading is given in terms of the root mean-square deviation:

∆g = h( g − hgi )2i p The relation between the (normalised) position and momentum wave function is given by the following formulae:

1 i ψ(r,t) = dp ϕ(p,t)e ~ p·r (2π~)3/2 Z 1 i ϕ(p,t) = dr ψ(r,t)e− ~ p·r (2π~)3/2 Z

Note that these wave functions describe the same state of the particle, the diﬀerence residing only in the fact that they refer to diﬀerent representations of the same state. Using the De Broglie relation p = ~k it is observed that ~3/2ϕ(~k,t) amounts precisely to a three-fold Fourier transform, in all three components of the position vector, of ψ(r,t) (see Appendix A on Fourier integrals). From the Fourier theorem we have the equality

dr |ψ(r,t)|2 = dp |ϕ(p,t)|2 Z Z by applying (a threefold) Parseval’s formula, which tells us that total probability is the same in either representation.

Observables as operators: momentum and position operator

We have seen that if we work in the position representation expectation values of observables that depend only on position can be easily calculated doing an integral over all space and including within the integral a multiplicative factor f(r) which represents the observable in question. Likewise, if we work in the momentum representation observables that depend only on momentum can be calculated including a multiplicative factor g(p) within the integral over p. However, if we would be interested in calculating mixed observables, depending on both r and p we have to ﬁnd a way how to represent momentum in the position representation, or vice versa how to describe postion within the momentum

6 representation. This is achieved by representing observables by means of operators acting on the wave function in question. This is again one of the key ideas in QM: observables are represented by operators acting on the Hilbert space of wave functions. This new point of view requires that we make a distinction between observables as operators and the numerical value that an observable might take on. We will make this distinction clear by equipping an operator by a hat: A denotes the operator associated with the observable A. The mathematical form that an b operator takes depends on the choice of representation in which we work. For instance, if we work in the position representation (i.e. using wave functions ψ(r,t)) the observable corresponding to the x,y,z-components of the position is just the operator of multiplication by the numerical value of the corresponding component of the position:

xψ(r,t) = xψ(r,t) yψ(r,t) = yψ(r,t) b zψ(r,t) = zψ(r,t) b where the left hand side should beb read as: the operator x resp. y, z acting on the wave function ψ(r,t), and the right habd side is just multiplication by the value corresponding to the argument of the function ψ(r,t) = ψ(x,y,z; t). If,b however,b b we work in the the momentum representation, position can no longer be represented in this simple way and is actually replaced by diﬀerential operators as follows ∂ xϕ(p,t) = i~ ϕ(p,t) ∂px b ∂ yϕ(p,t) = i~ ϕ(p,t) ∂py b ∂ zϕ(p,t) = i~ ϕ(p,t) ∂pz b Thus, in the momentum representation the position vector is represented by the vector diﬀerential operator which is (modulo the factor i~) the gradient in the space of momentum: ∂ r i~ ∂p A similar story holds for the momentumb operator: if we work in the momentum representation (i.e. using ϕ(p,t)) the momentum operator is simply the operator of multiplication by the numerical value of the components of momentum in the argument of the wave function, i.e.

pxϕ(p,t) = pxϕ(p,t) p ϕ(p,t) = p ϕ(p,t) by y p ϕ(p,t) = p ϕ(p,t) bz z but in the position representationb the momentum operator is represented as again a dif-

7 ferential operator ∂ p ψ(r,t) = −i~ ψ(r,t) x ∂x ∂ pb ψ(r,t) = −i~ ψ(r,t) y ∂y b ∂ p ψ(r,t) = −i~ ψ(r,t) z ∂z which amounts to saying thatb in the position representation the momentum vector is represented by the diﬀerential operator ∂ p −i~ ∂r which is proportional to the gradientb in terms of the position vector. One of the most striking consequences of this picture is the following: the operators representing the corresponding components of position and momentum do not commute! In fact, focusing for a moment on the x-components, it is easy to verify that: ∂ on the one hand : p xψ = p xψ = −i~ (xψ) x x ∂x ∂ψ b b = −b i~ψ − i~x ∂x ∂ψ ∂ψ on the other hand : xp ψ = x −i~ = −i~x x ∂x ∂x ⇒ (xpx − pxx)ψ = [x, px]ψ =bbi~ψ b where [·, ·] denotesbb the commutatorb b b bracketb of operators

[A, B]= AB − BA for any two operators A and B. Thus,b b we ﬁndb b theb operatorb identities b b [x, px] = [y, py] = [z, pz]= i~ where the right-hand side shouldb beb readb asb i~1 withb b 1 being the identity operator acting trivially. All other commutator brackets vanish: b b [x, y] = [x, z] = [y, z] = 0 [p , p ] = [p , p ] = [p , p ] = 0 bx b y b bx z b b y z We make the following remarks:b b b b b b

• The non-commutativity of the operators reﬂects the incommensurability of the corresponding observables: two observables are incomensurable iﬀ their corresponding operators do not commute. This gives us an algebraic method of deciding whether or not we are dealing with commensurable observables.

8 • The collection of all operators, subject to the commutation relations given above, forms an algebraic object in its own right which is called a Heisenberg algebra. The notion of an algebra (not to be confused with Algebra as a subject in Mathematics) will be given in Appendix C.

• The system of commutation relations does not depend on what representation we choose: the commutation relations remain the same whether we work in the position representation or whether we work in the momentum representation. As such the Heisenberg algebra is in a sense the more fundamental object in the theory.

Operators and Expectation values

Now that we know how to represent observables in either representation we can also calculate expectation values of such observables in any representation. This works as follows: suppose we want to evaluate expectation values of observables g(p) depending on momentum but in terms of the position wave function ψ(r,t) we must use the relation between ψ and ϕ to evaluate: hg(p)i = dp g(p)|ϕ(p,t)|2 Z 1 ~ 1 ′ ~ = dp g(p) dr ψ(r,t)e−ip·r/ dr′ψ∗(r′,t)eip·r / Z (2π~)3/2 Z (2π~)3/2 Z 1 ~ ′ ~ = dp dr ψ(r,t)e−ip·r/ dr′ψ∗(r′,t)g(−i~∇′) eip·r / (2π~)3 Z Z Z 1 ′ ~ = dr′ dr ψ∗(r′,t)ψ(r,t) g(−i~∇′) dp eip·(r −r)/ Z Z (2π~)3 Z = dr′ψ∗(r′,t)g(−i~∇′) dr δ(r′ − r)ψ(r,t) Z Z = dr′ψ∗(r′,t)g(−i~∇′) ψ(r′,t) Z where ∇′ = ∂/∂r′ and we have used the fact that each −i~∇′ acting on the exponent exp(ip · r′/~) produces a multiplicative factor p within the argument of the function g. Furthermore, we have used the three-dimensional δ-function

1 ′ ~ δ(r′ − r)= dp eip·(r −r)/ = δ(x′ − x) δ(y′ − y) δ(z′ − z) (2π~)3 Z Thus, we have the expression for the expectation value:

hg(p)i = dr ψ∗(r,t)g(−i~∇) ψ(r,t) Z in terms of the position wave function, noting that within the integrand we have the momentum operator entering via g(p).

9 In particular we have

hpi = dr ψ∗(r,t) (−i~∇) ψ(r,t) Z = dr (i~∇ψ∗(r,t)) ψ(r,t)= hpi∗ Z after integration by parts (the boundary term arising from the integration by parts vanishes in view of the fact that ψ tends to zero as r tends to inﬁnity). This demontrates that the expectation value is a real-valued quantity as we expect from any physical quantity. This is related to the fact that the momentum operator −i~∇ is a Hermitian (self-adjoint) operator, cf. Appendix C.

Equations of motion for Expectation Values

We are now in a position to calculate the time evolution of the expectation values. Note that both hri and hpi do not depend on r nor p (since we integrate over these) but only on time t. Thus, calculating d d ∂ hri = dr rP (r,t)= dr r P (r,t)= − dr r∇ · j dt dt Z Z ∂t Z where j is the probability current as before. Performing an integration by parts and using the identity (a · ∇)r = a we obtain

d hri = dr [−∇ (·j r)+ j]= j dr dt Z Z 1 1 = dr [ψ∗(−i~∇ψ) + (i~∇ψ∗)ψ]= hpi 2m Z m where we have used once again the fact that the integral over the divergence vanishes (as it produces a boundary term) and where we have inserted the explicit expression for j. The two terms in the last integral give the same contribution which accounts for the removal of the factor 1/2. Thus, we see that we get the usual (classical) relation between velocity vector and momentum, but of course only in terms of the expectation values! A similar calculation yields the equation for the time-evolution of hpi, namely

d d hpi = dr ψ∗(−i~∇ψ) dt dt Z 1 ~2 = dr ψ∗(−i~∇) − ∇2ψ + V ψ Z i~ 2m 1 ~2 − dr − ∇2ψ∗ + V ψ∗ (−i~∇ψ) Z i~ 2m ~2 = dr ψ∗∇(∇2ψ) − (∇2ψ∗) ∇ψ 2m Z − dr (ψ∗∇(V ψ) − V ψ∗∇ψ) Z

10 The integrand in the ﬁrst term can be written as a divergence: for instance the ith component of the integrand (which, remember, is a vector) can be written as

∗ 2 2 ∗ ∗ ∗ ψ ∇i(∇ ψ) − (∇ ψ ) ∇iψ = ∇ · [ψ ∇i(∇ψ) − (∇ψ ) ∇iψ] thus leading as before to a boundary term after integration, which we will assume to vanish. The second term yields: d hpi = − dr ψ∗(∇V )ψ = −h∇V i dt Z since V does not depend on p. Thus we get the classical equation of the motion in terms of the expectation value of the force −∇V deriving from the potential V (r,t). To summarise: the equations for the expectation values of position and momentum follow precisely the classical equations of the motion d 1 d hri = hpi , hpi = −h∇V i dt m dt which demontrates the correspondence principle (see Handout # 2): on the macroscopic level we observe only the motion of the averaged physical quantities which corresponds precisely to what we expect from the classical theory. This result is sometimes referred to as Ehrenfest theorem.

Mixed operators: Hamiltonian operator & Angular Momentum

The fact that physical observables are represented as operators acting on the wave function (in any representation) allows us to introduce operators for other physical observables. We know already how to represent momnetum and position in both representations. Inspired by the formulae from classical mechanics (see Handout # 1) we can now also introduce operators representing angular momentum and energy, the latter leading to the important Hamiltonian operator. These operators are often mixed in that they involve both momentum and position. When dealing with mixed observables h(r, p) one difficulty that arises is the one of the ordering of operators. In fact, replacing r and p simply by operators r and p leads to the difficulty that since these operators do not commute we have to invent a recipe for 2 2 b b2 2 putting them in some order. For example, the operator pxx is not the same as x px or as pxxpxx, etc. In principle, this is a rather delicate matter and has given rise to some theoretical considerations during the development of QM,b wbhich we will not go into.b b We willb justbb b assume that for the operators we are dealing with this problem can somehow be avoided. Two important operators that arise are the Hamiltonian operator, and the operator of angular momentum. The Hamiltonian operator is the quantum-mechanical counterpart of the Hamiltonian of the classical theory: it is the operator that represents the observable of energy. For the simple case of a single non-interacting particle subject to a potential field V (r,t) the Hamiltonian operator is simply given by: 1 Hamiltonian operator : H = p2 + V (r,t) 2m b b b 11 and there is no operator ordering ambiguity because the momentum and position dependent parts appear in separate terms3. When working in the position representation, replacing the momentum operator by −i~∇, we immediately recognize the operator ap- pearing on the r.h.s. of the Schrödinger equation. Thus, the SE can be recast in the operator form: ∂ Schrödinger Equation : i~ ψ = Hψ ∂t If the potential is time-independent, we can formally “integrate”b this equation by the formula: i ψ(r,t)= e− ~ Htψ(r, 0) = U(t)ψ(r, 0) b where the exponential of an operator is defined by theb formal series:

∞ 1 eA = An b n! nX=0 b The operator U is called the unitary operator of the time-evolution. Since H is a Hermitian operator, H = H† (see Appendix C), the operator U is a unitary operator: it obeys b b b b UU † = U †U = b1

1 being as before the identity (unit)b operator.b b b Inb general U is not easy to calculate in any explicit form and, thus, these expressions involving this operators will only be used to b b make some formal statements. in any case the Hamiltonian H is one of the most important objects in the theory, and most of the actual calculations in QM are dealing with the study b of this operator for the various systems that we are going to look at. The angular momentum operator is another important operator. It is the quantum- mechanical counterpart of the angular momentum vector ℓ = r × p. Its operator components are given by:

ℓx = ypz − zpy , ℓy = zpx − xpz , ℓz = xpy − ypx

This operator willb be investigatedbb bb moreb extensivelybb bb whenb webbwillb alsob discuss particles with spin. For the time being it suﬃces to recognize that the components of the vector angular momentum operator ℓ = r×p obey some nontrivial comutation relations amongst eachother, namely: b b b

[ℓx, ℓy]= i~ℓz , [ℓy, ℓz]= i~ℓx , [ℓz, ℓx]= i~ℓy b b b b b b b b b

Exercise: Prove these commutation relations on the basis of the deﬁnitions of the ℓx, ℓ , ℓ and using the Heisenberg commutation relations between the components of the y z b position and momentum operators. b b 3In more complicated quantum-mechanical systems the operator ordering ambiguity might play a role.

12 Uncertainty relations & Minimal wave packet

We are now in a position to prove the Heisenberg uncertainty relations: for every state of the particle we have ~ ~ ~ ∆x ∆p ≥ , ∆y ∆p ≥ , ∆z ∆p ≥ x 2 y 2 z 2 In Handout # 2 we have given relations of this form without having a precise criterium on how to measure the spreading. Here, the spreading is measured by the root mean- square deviation, so we can actually prove the uncertainty relation and make more precise statements than before. Thus, we have for example:

hxi = dr x|ψ|2 , (∆x)2 = dr (x − hxi)2|ψ|2 Z Z The main tool in the proof is the Cauchy-Schwarz (CS) inequality (see Appendix C for a proof under very general circumstances):

2 2 2 ∗ dr |ψ1| dr |ψ2| ≥ dr ψ ψ2 Z Z Z 1

for square-integrable functions ψ1, ψ2. The equality sign holds if and only if (assuming the function ψ2 is not identically zero) ∗ dr ψ2ψ1 ψ1 = λψ2 where λ = 2 R dr |ψ2| R Proof: We will concentrate on proving the uncertainty relation for the x-components. The other uncertainty relations follow by analogy. Let us apply the CS inequality to the functions ∂ ψ = (x − hxi)ψ , ψ = i~ + hp i ψ 1 2 ∂x x where ψ is the wave function of our system. Obviously,

2 2 dr |ψ1| = (∆x) Z Furthermore,

2 ∂ ∗ dr |ψ2| = dr −i~ + hpxi ψ ψ2 Z Z ∂x ∞ ∞ ~ ∗ ∞ ∗ ~ ∂ = dy dz (−i ψ ψ2)|x=−∞ + dr ψ i + hpxi ψ2 Z−∞ Z−∞ Z ∂x 2 ∗ ∂ 2 = dr ψ −i~ − hpxi ψ = (∆px) Z ∂x where in the ﬁrst step we have done an integration by parts and where we assume the boundary term to vanish as ψ → 0 as x → ±∞. Subtracting

∗ ∗ ∂ dr ψ ψ2 = dr (x − hxi)ψ i~ + hpxi ψ Z 1 Z ∂x

13 from

∗ ∂ ∗ dr ψ ψ1 = dr −i~ + hpxi ψ (x − hxi)ψ Z 2 Z ∂x

∗ ∂ = dr ψ i~ + hpxi (x − hxi)ψ Z ∂x

∗ ∗ ∂ = i~ dr ψ ψ + dr ψ (x − hxi) i~ + hpxi ψ Z Z ∂x where in the successive steps we have done an integration by parts, and used the fact that moving the operators i~(∂/∂x)+ hpxi and x − hxi through eachother prodices an extra terms i~ (which accounts for the extra term on the right-hand side), we obtain:

∗ ∗ ∗ i~ = dr ψ ψ1 − dr ψ ψ2 = 2i Im dr ψ ψ1 Z 2 Z 1 Z 2 Putting all the ingredients together we get now the string of inequalities

~ 2 2 2 ∗ ∗ 2 2 = Im dr ψ ψ1 ≤ dr ψ ψ1 ≤ dr |ψ1| dr |ψ2| 2 Z 2 Z 2 Z Z

(the ﬁrst inequality following from the fact that for any complex number z we have Imz ≤ |z|) which yields immediately the inequality

~ 2 ≤ (∆x)2(∆p )2 2 x from which the uncertainty relation follows.

One may ask the question for what wave function ψ we can actually achieve the equal sign in the inequality, thus leading to a “minimal” wave packet (i.e. one with minimal uncertainty). The equal sign is achieved iﬀ two conditions are met:

• ψ1 = λψ2 with λ given above to satisfy the equality in the CS inequality, and

∗ • Re dr ψ2ψ1 = 0 , i.e. the condition under which the square of the imginary part is equalR to the modulus squared.

The second condition, tells us that λ is purely imaginary, so λ = iλ0 with λ0 real. Inserting the expressions for ψ1, ψ2 into the first condition gives us a first order differential equation for ψ, namely ∂ (x − hxi)ψ = iλ i~ + hp i ψ 0 ∂x x Solving this (by separation of variables) gives us the solution of the form:

i 1 2 ψ(x)= C exp hpxix − (x − hxi) ~ 2~λ0

14 where C is some integration constant (independent of x). Using the expression obtained ∗ ~ above for Im ψ2ψ1 dr = /2, and inserting the condition ψ1 = λψ2 we can express λ0 2 in terms of theR spreading ∆x, namely we get: λ0 = 2(∆x) /~ . Furthermore, C can be speciﬁed by normalising the wave packet, and thus we obtain ﬁnally the following formula for the Gaussian minimal wave packet:

2 1 1 i hp ix− x−hxi ψ(x)= e ~ x 4 “ ∆x ” (2π(∆x)2)1/4

There is no time-dependence in this formula because as soon as the time is switched on (i.e. the wave packet evolves according to the Schr¨odinger equation) the wave packet starts to spread out and will no longer be one with minimal uncertainty. Remark: Note that we can, by similar methods, also ﬁnd the minimal wave packet in the momentum representation. This leads to the result:

2 1 1 − i hxip− p−hpi ϕ(p)= e ~ 4 “ ∆p ” (2π(∆p)2)1/4 where p = px.