Understanding the Schrodinger equation as a kinematic statement: A probability-first approach to quantum
UUnderstanding the Schr¨odinger equation as a kinematic statement 1
Chapter 1 U NDERSTANDING THE S CHR ¨ ODINGEREQUATION AS A KINEMATIC STATEMENT : A
PROBABILITY - FIRST APPROACH TO QUANTUM
James Daniel Whitfield ∗ Department of Physics and AstronomyDartmouth CollegeHanover, NH, US
Abstract
Quantum technology is seeing a remarkable explosion in interest dueto a wave of successful commercial technology. As a wider array of engi-neers and scientists are needed, it is time we rethink quantum educationalparadigms. Current approaches often start from classical physics, linear al-gebra, or differential equations. This chapter advocates for beginning withprobability theory. In the approach outlined in this chapter, there is less in theway of explicit axioms of quantum mechanics. Instead the historically prob-lematic measurement axiom is inherited from probability theory where manyphilosophical debates remain. Although not a typical route in introductorymaterial, this route is nonetheless a standard vantage on quantum mechan-ics. This chapter outlines an elementary route to arrive at the Schr¨odingerequation by considering allowable transformations of quantum probabilityfunctions (density matrices). The central tenet of this chapter is that proba-bility theory provides the best conceptual and mathematical foundations forintroducing the quantum sciences. ∗ E-mail address: james.d.whitfi[email protected] a r X i v : . [ phy s i c s . pop - ph ] M a r A probability-first approach to quantum
1. Introduction
This chapter is meant to give a new pedagogical paradigm for understanding quan-tum mechanics as an extension of probability theory. Viewing quantum this way isnot at all new, but it is rarely put as the central tenet of introductions to quantumtheory. Instead, quantum theory is usually introduced as an extension of classicalmechanics as is done in some of the other chapters of this book. The purpose ofthis chapter is to provide an alternative to traditional methods for understandingthe Schrodinger equation starting from probability rather than classical mechan-ics. This chapter will outline the approach and its advantages when explaining theSchr¨odinger equation.The approach taken here does not appeal to historical derivations of theSchr¨odinger equation nor has it been tied to classical mechanics (e.g. Hamilto-nian methods and Poisson brackets). Deriving the Schr¨odinger equation this wayavoids discussing the physical interpretation of the Hamiltonian and, more gen-erally, of energy. I put this forward as an advantage to this approach. Energy isa difficult concept for the uninitiated and deserves its own full-fledged discussionwhich can be postponed until after the introduction of quantum theory dependingon the composition of the audience. By starting with a wide variety of examplesfrom probability, introductions to quantum theory made in physics, engineering,philosophy, mathematics or computer science from a common starting point. Withthe proliferation of quantum technology, not all students of quantum theory arephysicists and may be easily confused by the energy concept.Moreover, in this approach, there are very few postulates explicitly required forquantum theory. For instance, there are no explicit postulates governing evolutionnor measurement. The former is a result of kinematic constraints on the set ofvalid states. The latter is implicitly postulated by choices made within probabilitytheory. This way quantum measurement, its interpretation, its consequences uponrealization are all imported directly from probability theory.Kinematics is the study of the motion of objects without reference to the causeof that motion. When teaching introductory mechanics, before momentum, beforeenergy, and even before forces, modern pedagogy begins with kinematics. Thisidea can equally be applied to probability theory. The kinematic understandingof probability theory can be nearly directly bootstrapped to quantum theory. Thischapter is an exposition advocating for this approach to be adopted for training thenext wave of scientists and engineers on the essentials of quantum theory.After thoughtful consideration, I believe serious practitioners of quantum the-ory will recognize all the arguments found here and will likely not have any majordisagreement. However, the immediate introduction of the quantum density matrixis new to introductory pedagogy. Due to the newness, exercises and elementarynderstanding the Schr¨odinger equation as a kinematic statement 3textbook references are not directly available and would need to be adapted to thismethod of presenting quantum. An effort has been made to define all terms for thebeginner, but the emphasis of the chapter is for instructors and mentors who needto teach quantum to newcomers.
2. Probability theory
The journey toward the Schr¨odinger equation starts not with linear algebra norcalculus but with probability theory. Probability theory can be understood and pre-sented at a variety of levels depending on the target audience and skills or interestsof the student. This section and the next are included in the chapter to provide thelogical basis for quantum theory. The sections on probability theory are primar-ily to establish notation and correct understanding of quantum theory. However,in principle, they could be streamlined if the audience possesses sufficient back-ground in mathematics.A key asset of the probability-first approach to quantum is that measurementcan be introduced before discussing quantum theory. While this does not resolvenor curtail discussion of quantum foundations, it makes clear that the majority ofthese issues and mysteries of measurement are largely within the domain of proba-bility theory. This allows learners to focus on what is new within the quantum ex-tension of probability without being confused or misled concerning the philosophyof quantum mechanics. Additionally, it allows quantum theory to be introducedwithout a separate “measurement axiom.”Only the basics of probability theory are needed to arrive at a conceptual un-derstanding of the Schr¨odinger equation. Depending on the application areas ofquantum theory planned, more or less time may be dedicated to probability theoryand examples that can be revisited in the quantum domain. For a shorter, moreintuitive discussion of probability theory may allow linear algebra to be introducedand developed. However, a more formal introduction is appropriate for practition-ers interested in reading and contributing to the mathematical areas of quantumresearch.Regardless of the depth of the discussion of probability, certain concepts shouldbe introduced at this level given the intuition that can be exposed using probabilitytheory. This helps give more concrete examples from probability theory when dis-cussing mathematics. The choice of examples can be concerted with the examplesthat will be introduced later in quantum theory.To extend probability theory to quantum theory, the notion of an N dimensionalorthonormal vector space basis needs to be introduced. Then each of N elementaryevents can be associated with the N basis vectors. Introducing a probability vector A probability-first approach to quantumeasily allows discussing vector spaces, norms, and inner products . At minimum,the notation used for these concepts should be fixed at this point.The choice of notation in elementary texts is split between Dirac notation andmore standard mathematical notation. Those coming from a mathematical back-ground may be confused by some of the choices of Dirac notation. On one hand,since its use is nearly exclusively within quantum theory, Dirac notation requires alonger discussion regardless of mathematical preparation. On the other hand, intro-ducing Dirac notation immediately, allows learners to access more of the modernliterature faster. Further, the early introduction at the level of probability theoryallows for many exercises before moving to quantum theory.In this article, we utilize Dirac notation. Briefly, in Dirac notation we denotevectors as (cid:126)x = | x (cid:105) and the conjugate transpose of a vector as (cid:104) x | = ( (cid:126)x ∗ ) T = (cid:126)x † .The inner product most students have been exposed to is the dot product of vectors: (cid:126)x.(cid:126)y = ( (cid:126)x ) † (cid:126)y = (cid:80) j x ∗ j y j = ( (cid:126)y.(cid:126)x ) ∗ . In terms of Dirac notation, one writes (cid:126)x.(cid:126)y = (cid:104) x | y (cid:105) .If we want (cid:104) y | Ax (cid:105) to be equal to the inner product of some operator B actingin the dual space such that (cid:104) By | x (cid:105) = (cid:104) y | Ax (cid:105) , then B is called the adjoint of A .In standard notation, the adjoint of A is denoted A † . In the finite spaces we arediscussing here, A † is the conjugate transpose of matrix A .Even with more advanced audiences, it remains a good idea to explicitly definethe conjugate of a complex number as z ∗ = ( a + bi ) ∗ = ( a − bi ) with i = √− andthe adjoint of a matrix as ( A † ) mn = A ∗ nm . In mathematical literature, this notationfor adjoint and conjugate operators is often reversed.The pedagogical development towards quantum theory begins with introduc-ing probability distributions as vectors. Suppose we have an experiment whoseoutcomes depend on chance. The sample space of the experiment, Ω , is the set ofall possible outcomes. In this chapter, we consider sample spaces with N discreteexclusive outcomes. Depending on the emphasis and purposes of an introductionto quantum theory, this sample space may be considered continuous or discrete (fi-nite or countably infinite). Each of the N elementary events is associated with For a course discussing common examples such as the hydrogen atom and real-space wavefunctions, the continuous formulation may be introduced here. Then the probability distribution isgiven by a function p ( x ) where Pr( X = x ) = p ( x ) . The normalization is also unity over the samplespace; however, this is now defined by an integral: (cid:82) Ω p ( x ) dx = 1 . Similarly, (cid:82) ba p ( x ) dx = Pr( a ≤ X ≤ b ) where a, b ∈ Ω . The L p norms are defined as | f | p = ( (cid:82) | f ( x ) | p dx )) /p . nderstanding the Schr¨odinger equation as a kinematic statement 5orthonormal vectors {| e j (cid:105)} Nj =1 . Then, one can write a probability vector as | p (cid:105) = (cid:88) j p j | e j (cid:105) (1)The numerical components of the probability vector are given by p j = (cid:104) e j | p (cid:105) .For use within probability theory, each p j satisfies ≤ p j ≤ and satisfy thenormalization condition (cid:80) j p j = 1 . Its important to highlight that {| e j (cid:105)} is fixedbut otherwise arbitrary. Obtaining the outcome of an experiment is a measurement . The actual act of mea-suring the outcome of an experimental realization is not necessary for introduc-ing quantum theory. By maintaining a conceptual link to probability theory whenapproaching quantum theory, connecting experiments and outcomes remains anexercise in probability theory. Then questions such as assigning, updating, andmeasuring a probability distribution before, after or even during a measurementare the same in quantum theory. We make the argument clearer when discussingcoherence and decoherence of quantum states.In standard treatments of quantum theory, measurement is often tacked on asa postulate of quantum theory. However, with quantum theory as an extensionof probability theory the measurement postulate is obtained from ordinary prob-ability theory. By examining the quantum kinematic constraints, we will obtainthe Schr¨odinger equation. Before moving on to quantum generalizations, the dis-cussion of kinematic constraints can be prepared by examining the kinematics ofprobability distribution functions.With the definitions and requirements of probability distributions, we can con-sider the possible transformations and evolutions of probability distributions with-out reference to the causes of these changes. It is this agnosticism that will helpexpose a broad understanding of quantum theory but may introduce difficulties inconnecting to applications. This can be ameliorated by choosing examples andillustrations appropriate for the target audience. Orthonormality requires that the set is both mutually orthogonal (i.e. (cid:104) e j | e k (cid:105) = 0 if j (cid:54) = k ) andindividually normalized under the L norm induced by the inner product (i.e. (cid:104) e j | e j (cid:105) = 1 ). Theorthogonality corresponds to the exclusivity of the elementary event such that if event j occurs, thenevent k did not for all k (cid:54) = j . A probability-first approach to quantum
3. Kinematics of probability distributions
Schr¨odinger’s equation and all other quantum equations of motion must obey kine-matic constraints that ensure the form of the quantum state remains valid. Thiskinematic approach is first illustrated using probability vectors in this section,which is later generalized to quantum theory in 5..In this section, we wish to consider transformations that take one valid prob-ability distribution | p (cid:105) to another valid probability distribution | p (cid:48) (cid:105) . We considerseveral classes of transformations that provide direct parallels to the quantum kine-matic discussion. The change of basis concept is essential to the development ofthe Schr¨odinger equation as a kinematic expression. Also included in this sectionis a discussion of the master equation describing differential transformations ofprobability distributions. The quantum extension of this differential equation goesbeyond the Schr¨odinger equation and, thus, is outside the scope of this chapter.However, in the context of mentoring or teaching, this provides a nice way to openthe discussion of stochastic quantum processes and general evolution equations forquantum systems in contact with the environment.The argument of this chapter requires viewing the change of basis formula forprobability vectors as just an exchange of labels. The arbitrary map from elemen-tary events to basis vectors allows us to rearrange the vector without changing anyof the values by relabelling the event or relabelling the basis vectors. This is doneusing a permutation P ∈ S m where S m is the symmetric group of permutations of m objects.If we consider each permutation as an m × m matrix, then it is easy to intro-duce matrix multiplication exercises using Dirac notation. Simple manipulationssuch as p (cid:48) j = (cid:88) k P jk p k (2) (cid:104) e j | p (cid:48) (cid:105) = (cid:104) e j | ( P | p (cid:105) ) (3) = (cid:88) k (cid:104) e j | P | e k (cid:105)(cid:104) e k | p (cid:105) (4)are useful for more elementary audiences. Here, it is also useful to introduce theresolution of the identity = (cid:80) j | e j (cid:105)(cid:104) e j | and emphasize that it holds whenever { e j } is an orthonormal basis. This can help simplify later discussion and clarifynotations for more advanced students unfamiliar with Dirac notation. For example, a representation of S is given by P = , P = and the matrix products P P , P P , P P , P P P . nderstanding the Schr¨odinger equation as a kinematic statement 7Stochastic matrices are introduced via convex combinations of permutationmatrices in this section. This is not the only way to introduce transformations thatpreserve the kinematics constraints of probability vector transformations. More-over, we do not reach the full class of stochastic matrices but rather bistochastictransformations. For the purposes of reach the Schr¨odinger equation, it is not nec-essary to go beyond bistochastic matrices.To get a transformation that preserves the validity of probability vectors, sup-pose with probability w j permutation P j is applied ( (cid:80) j w j = 1 ). Then the trans-formation M = (cid:88) j w j P j (5)can act on | p (cid:105) giving M | p (cid:105) = | p (cid:48) (cid:105) as a valid probability distribution. Matrices suchas M that transforms valid probability distributions to valid probability distribu-tions are called called stochastic matrices . The use of the values M ij as transitionprobabilities is a useful point of reference before moving to quantum theory M ij = Pr( j → i ) = Pr( X (cid:48) = i | X = j ) = Pr( X (cid:48) = i and X = j )Pr( X = j ) (6)This serves as a way to introduce conditional probabilities P ( A | B ) and give awealth of examples to illustrate matrix multiplication in a more concrete fashionwith transition matrices.Since the Schr¨odinger equation determines differential changes, it is useful todescribe valid transforms of probability distributions via a differential equation.The typical way to characterize the infinitesimal changes of a probability densityfunction is in terms of a master equation. In such equations, the rate of changein one component of the probability vector is the rate of probability entering thatcomponent minus the rate of probability decreasing from that component. With the This construction of stochastic matrix M as a convex combination of permutation matrices isa way to characterize all bistochastic matrices [2]. More general one-sided stochastic matrices arepossible and are the reason that (cid:80) j M ji is not necessarily unity in (8). Many probability books suchas [1] prefer the stochastic matrices written such that ˜ M ij = Pr( i → j ) . In this case, probabilityvectors are row vectors x multiplying from the left x ˜ M rather than column vectors multiplying fromthe right M | p (cid:105) . The bistochastic matrices can work with probability vectors as column vectors onthe right or row vectors on the left; hence the prefix ‘bi-’. Here, we only consider probability vectorsas column vectors. A probability-first approach to quantumrate of going from state i to state j as M ij , then in matrix form, we writecomponentchange = [ rate in ] − [ rate out ] (7) dp k = (cid:88) j M ij p j − (cid:88) j M ji p i (8)We can further summarize this statement by first defining the degree matrix as D kk = (cid:80) j M jk and then the generator of the evolution is given by L = M − D such that L | p (cid:105) = d | p (cid:105) dt (9)The real variable t is used to parameterize the evolution under L . For simplicity,we assume that the matrix L is time-independent.We can integrate the equation following L | p (cid:105) = d | p (cid:105) dt = ⇒ Ldt = d | p (cid:105)| p (cid:105) (10) = ⇒ Lt = log | p t (cid:105) − log | p (cid:105) (11) = ⇒ e Lt = exp(log | p t (cid:105) − log | p (cid:105) ) (12) = ⇒ e Lt | p (cid:105) = | p t (cid:105) (13)For all positive real values of t , M ( t ) = exp( Lt ) , is a stochastic matrix. Theanalog of (13) in quantum theory is beyond the Schr¨odinger equation; however,its integration is instructive. The formulation of (9) has much more in commonwith the Schr¨odinger equation than the standard equations found in a first courseon classical mechanics. From a pedagogical perspective, the derivation also offersan opportunity to introduce more mathematical concepts including integration andthe matrix exponential.If readers are unfamiliar with linear algbera, it might be useful to introducematrix mechanics using a programming environment (e.g. Python). The numeri-cal tools built or utilized here should be planned to have their quantum extensionsdiscussed. This allows students to test and play with the concepts and equations onsmall numerical examples before reaching quantum theory. The eigendecomposi-tion, Taylor expansion, and matrix functions can be introduced alongside (13).
4. Quantum theory via probability
With the sufficient notions and definitions from probability theory and linear alge-bra established, we can proceed to discussing quantum theory as a generalization ofnderstanding the Schr¨odinger equation as a kinematic statement 9Figure 1. A graphical illustration of the density matrix. The diagonal elementsform a valid probability distribution and give the propensity for an outcome to berealized upon measurement in the depicted basis. The off-diagonal element arecalled coherences. This illustration is particularly useful for elementary discus-sions with less mathematical involvement.probability theory. The quantum probability density matrix in quantum mechanicsis the direct generalization of the probability distribution vector.Quantum probability distributions, often referred to as the density operator ordensity matrix, are commonly denoted with ρ . Valid density matrices must sat-isfy the normalization condition (cid:80) i ρ ii = 1 which replaces the L condition ofprobability vectors. The probability vector consists of only positive numbers. Thecomparable statement for density matrices is that for every normalized (possiblycomplex) vector (cid:126)v , the expectation value (cid:104) v | ρ | v (cid:105) is real and greater or equal to zero(see Fig. 1). Some introductory material on quantum theory suggests that the wave functionis analogous to the probability vector with the key difference being the L normis used for the former and the L norm for the latter. In the probability first ap-proach, the L norm remains applicable to the diagonal of the density matrix. Theoff-diagonal elements are called coherences and are strictly quantum. The develop-ment of coherences is what allows for a separation between quantum and ordinaryprobability theory.The density matrices formed by taking a probability vector and converting to amatrix with the same probability values along the diagonal (i.e. ρ = Λ (cid:126)p = diag ( (cid:126)p ) )with zeros elsewhere. See Fig. 2 for an example of the correspondence. The set ofmatrices formed by taking probability vectors and placing their entries along thediagonal are all valid density matrices.Due to the conditions on valid density matrices, there always exists an or- Alternatively, this can be stated more succinctly: the density matrix is Hermitian, positive semi-definite, and has a normalized trace. natural basis of the quantum system since it is defined withoutexternal references. In the natural basis, say { ψ j } , there are no coherences and thedensity matrix can be written ρ = (cid:88) p j | ψ j (cid:105)(cid:104) ψ j | (14)with (cid:80) p j = 1 . The basis of the density matrix in Fig. 2 is the natural eigenba-sis and the diagonal elements give the probability for obtaining a particular basisvector upon measurement.If the reader notices, wave functions have not been mentioned explicitly. Foraudiences with some exposure to quantum theory, it is useful to connect wavefunctions back to the density matrix alongside their introduction. Then the densitymatrix in its eigenbasis can be described as a probability distribution over wavefunctions. The apt illustration of (14) is a wave function generator that outputsstate ψ j with probability p j .Another difference between a quantum density matrix and a probability vectorare the allowable changes to the basis. Previously, we allowed a change of basisdefined by a permutation matrix P . In quantum theory, we can now rotate the basiswithout changing the eigenvalues found in (14).Thus, quantum theory is introduced using a quantum probability density oper-ator that generalizes the probability distrubution vectors. We can now turn towardgeneralizing the possible change of basis allowed under quantum theory. This willlead us directly to the Schr¨odinger equation.nderstanding the Schr¨odinger equation as a kinematic statement 11
5. Transformations of the quantum probability densitydistributions
In this section, the allowable transformations of quantum density matrices is ap-proached in the same fashion as in section 3. We first illustrate the generalizationof probabilistic transforms in the quantum density matrix setting. Then the general-ization of the change of basis is introduced. Finally, the allowable transformationsof quantum density matrices are reached in direct parallel to the probability case.This development is logical and sensible for introducing quantum theory kine-matically. However, the generalization of stochastic matrices to the quantum do-main is more than is necessary for the Schr¨odinger equation. For the Schr¨odingerequation, it is enough to appreciate the continuous change of basis allowable fordensity matrices without composition.Earlier, permutations served as the change of basis for probability vectorswhere P (cid:126)p = (cid:126)p (cid:48) . To effect the same transformation on diagonal density matrix, Λ (cid:126)p , we use P Λ (cid:126)p P † = Λ (cid:126)p (cid:48) (15)Similarly, we have for M = (cid:80) w k P k the equivalent in terms of quantum densityoperators. (cid:88) k w k P k Λ (cid:126)p P † k = Λ M(cid:126)p (16)Now instead of just permuting the basis vectors, we may additionally rotate thebasis vectors from set { e j } of basis vectors to another { f j } . At this point, manyexamples can be used to illustrate the notion of differing basis used to describe thesame object.For a chemical audience, atomic orbital basis and the molecular orbital basisprovide an example of two bases for describing a single density matrix. In solidstate physics, the change of basis from k -space to the real space is an instruc-tive use of the Fourier transform. Examples in the lab frame and in the rotatingframe are appropriate if magnetic resonance is planned as part of the quantumcourse. Two-level systems (i.e. qubits) provide a wealth of illustration, examplesand demonstrations. The Bloch sphere and the Stern-Gerlach experiment are twoexamples that also help visualize the idea of multiple bases describing the sameobject. A simple example is found using plane rotations in two dimensions whichprovide mathematical and graphical illustrations and exercises.Regardless of the examples chosen, the change of basis is represented by amatrix that converts each orthonormal basis vector from one set { e j } Nj =1 to another2 A probability-first approach to quantumorthonormal set { f j } Nj =1 . This is arranged in matrix form as U = (cid:88) j | e j (cid:105)(cid:104) f j | = (cid:88) j ( (cid:126)f j )( (cid:126)e j ) † (17)In general, we know that U is a valid transformation of the basis if U † = U − .Such matrices are called unitary and pervade quantum theory.The change of basis is required in many settings. For example, the quantumstate may have been prepared in basis { e j } but the experiment is performed in basis { f j } . Then the probabilities determining the outcome of experiments is then givenby the diagonal of U Λ p U † with U given in (17).Similar to the characterization of stochastic matrices, we can characterize moregeneral transformations from one valid density matrix ρ to another ρ (cid:48) by changingthe initial basis with various probabilities. ρ (cid:48) = (cid:88) k w k U k ρU † k (18)This is the quantum generalization of equation (16). The arrival of equation (18) is quite beautiful in that it structurally echoes theprobability evolution in (16). Noting that permutation matrices are also unitarymatrices, the (16) reduces to a special case of (18). Here many asides could bemade depending on the audience and its aims. Noise, thermodynamics, entangle-ment, communication, and many other topics can be introduced once (18) has beenintroduced.A return to the discussion of measurement is also appropriate following (18). Itis sufficient to consider measurements as those of ordinary probability theory usingthe diagonal elements of the density matrix in the right basis. Decoherence theoryis also an instructive tangent where decoherence can be summarized as the loss ofthe coherences in the density matrix. With all the coherences zero, the density ma-trix is an ordinary probability distribution. A measurement channel is illustratedin (Fig. 3) but its usefulness in elementary courses can obscure the connectionto reality without sufficient examples. Connecting the mathematical formalism tophysical examples is an important exercise that can occasionally confound new-comers.In many ways, a natural direction continues with the kinematic derivation ofthe quantum analog of (8). Then the Schrodinger equation appears as the non-dissipative portion of the evolution. This is a choice that should be balanced with Similar to the footnote following (16), the expression in (18) is also not the most general. Thebroadest characterization of transformations from ρ to ρ (cid:48) is characterized by the Kraus representation: (cid:80) k E k ρE † k where the only constraint on the operators { E k } is that = (cid:80) E † k E k . This is clearlysatisfied for { E k = √ w k U k } as found in (18). nderstanding the Schr¨odinger equation as a kinematic statement 13Figure 3. A graphical illustration of the quantum aspects of measurement. Thequantum channel E meas transforms a density matrix to a probability distribution inthe basis of the measurement. Then the outcome realization and its consequencesare issues strictly within the domain of ordinary probability theory.the eventual goals of learning quantum theory. The lengthier discussion does notadd much, if it is unlikely to be used later in the course or in the research trajectoryof the students.With both the concept of the density matrix and its transforms as extensions ofprobability theory, we now turn to obtaining the Schr¨odinger equation as kinematicstatement.
6. The Schr ¨odinger equation
The structure of stochastic and quantum transformations are highlighted by theparallels of (16) and (18). The quantum master equation that generalizes (8) is un-necessary for the purposes of understanding the Schr¨odinger equation. It is enoughto consider the special case of (18).To arrive at the Schr¨odinger equation, we just need to find the infinitesimalbasis rotation of a specific quantum event | ψ (cid:105) that occurs with probability 1. Sucha quantum state is given by ρ = | ψ (cid:105)(cid:104) ψ | . Next, we parameterize the change fromone density matrix to another by a family of single unitary transformations U ( t ) where t represents time: ρ t = U ( t ) ρU ( t ) † (19) = U ( t ) | ψ (cid:105)(cid:104) ψ | U ( t ) † (20)Since the direct and dual spaces are redundant, we can simplify (20) as | ψ t (cid:105) = U ( t ) | ψ (cid:105) (21)For the final step, we will need an additional fact about unitary matrices that con-nects them with the matrix analog of a real number. First, one can consider a4 A probability-first approach to quantumminimal case where the unitary matrix consists of a single element. Then we cansatisfy condition uu ∗ = uu † = u ∗ u = with u = exp( − iθ ) as long as θ is real.To motivate the matrix equivalent, we define the matrix analog of the real partof a complex number. For arbitrary matrix A , the Hermitian part is ( A + A † ) inanalogy to scalar equation Re ( z ) =Re ( a + bi ) = ( z + z ∗ ) = a . For completeness,it may be worthwhile to continue the analogy by introducing skew-Hermitian ma-trices ( A − A † ) are the matrix analog of the imaginary part of a complex number.The property that H = H † is in analogy with a = a ∗ in the scalar case. Fromthere, the analog with u = exp( − iθ ) continues as U = exp( − iH ) = ∞ (cid:88) n =0 ( − iH ) n n ! (22)Some care must be given when taking the matrix exponential, in that one cannotexponentiate element-wise. This is most simply explained using eigendecomposi-tions or alternatively using the Taylor expansions as in righthand side of the finalequality of (22).Now to obtain the analog of (13), we still need to generate a family of unitarymatrices connected via a real-valued parameter. This is easily accomplished byextending (22) with real parameter tU ( t ) = exp( − iHt ) (23)Unlike the situation of stochastic evolution, this parameter can be positive or nega-tive. Thus, the sign convention of the last equation is arbitrary, but our conventionhere follows standard practices of quantum theory.Finally, we can take the derivative of (21) to arrive at d | ψ t (cid:105) dt = ddt U ( t ) | ψ (cid:105) (24) = − iHU ( t ) | ψ (cid:105) (25) d | ψ t (cid:105) dt = − iH | ψ t (cid:105) (26)Finally, we have arrived at the time-dependent Schr¨odinger equation in (26) with-out appeal to classical mechanics.It is worthwhile to point out that the probabilistic differential equation of mo-tion described earlier (8) is not analogous to the Schr¨odinger equation. This isbecause the Schr¨odinger equation describes how the rotation of the vector spacebasis occurs infinitesimally. Given that there is no continuous transformation be-tween discrete permutation matrices, there is no direct analog in probability theory.However, there is a direct quantum analog of (8) resulting in a differential form forthe transformations in (18) which is called the Lindblad equation.nderstanding the Schr¨odinger equation as a kinematic statement 15
7. Outlook
In the approach that this chapter advances, quantum theory is seen as an extensionof probability theory. Foundational issues such as the epistemic (knowledge) orontic (being) nature of the probability function can be addressed here before ap-proaching quantum theory. There are interesting foundational and philosophicalissues concerning the interpretation of probability theory which can be identifiedand set aside before discussing quantum theory. Depending on the tastes of a lec-turer or the learners, segues into the frequentist and inference interpretations ofprobability theory and of quantum theory can be explored in parallel.Regardless of what the reader or teacher chooses to pursue next, this chapterhas focused on understanding the Schr¨odinger equation from a kinematic pointof view. Introductory mechanics is taught at the high school level whereby allnecessary analytic tools are introduced as needed, and the course is even taughtwithout calculus. The prescription here aims to give an approach that allows acourse to be taught at various levels depending on the sophistication of the studentbut following a basic structure similar to introductory mechanics.Questions of prerequisites, necessary mathematical background, and whetherto begin with wave mechanics or discrete systems are answered as followed: noprerequisites should be required for an introductory course beyond a solid highschool education and an ambition to learn. To be sure, linear algebra, calculus,and differential equations are important for any serious study of physics or relatedtechnical subjects, but for an introduction to quantum theory, they are superfluousand can be introduced as needed.Depending on the nature of the course and the goals of the learner, there is awide variety of directions to continue following a probability-first introduction toquantum theory. Since the energy concept was not stressed in this presentation, it isan natural next step. Moreover, given that this presentation has already introducedprobability theory and the density matrix, open systems can be introduced imme-diately after the discussion of energy. Thermodynamics could also follow closelybehind.The notion of entropy and connected concepts in thermodynamics can be in-troduced alongside the probability introduction and re-examined in the quantumcontext. There are three sensible reasons to consider entropy directly after intro-ducing quantum theory: (i) it can be introduced at various levels directly usingprobability theory, (ii) historically, the idea of quantization has its origins in sta-tistical mechanic considerations (i.e. the ultraviolet catastrophe that led Planck tointroduce quantization as a concept), and (iii) entropy opens the door to discussionof entanglement entropy and algorithms for exploiting low entanglement [3].There may remain reluctance to switch given the overhead it places on craft-6 A probability-first approach to quantuming new notes, selecting new texts, and designing the correct scaffolding for vari-ous levels and backgrounds. However, I hope the elegance alone persuades someteachers and mentors to switch to a probability-first approach.
References [1] Grinstead, C. and Snell, J. L. (1997)
Introduction to Probability . Providence,RI: American Mathematical Society.[2] Horn, R. A. and Johnson, C. R. (2012)
Matrix Analysis . New York, NY: Cam-bridge University Press.[3] Verstraete, F. and Murg, V. and Cirac, J. (2008). Matrix product states, pro-jected entangled pair states, and variational renormalization group methods forquantum spin systems.