A completely algebraic solution of the simple harmonic oscillator M. Rushka, and J. K. Freericks
Citation: American Journal of Physics 88, 976 (2020); doi: 10.1119/10.0001702 View online: https://doi.org/10.1119/10.0001702 View Table of Contents: https://aapt.scitation.org/toc/ajp/88/11 Published by the American Association of Physics Teachers
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Landau levels, edge states, and gauge choice in 2D quantum dots American Journal of Physics 88, 986 (2020); https://doi.org/10.1119/10.0001703 A completely algebraic solution of the simple harmonic oscillator M. Rushkaa) and J. K. Freericksb) Department of Physics, Georgetown University, 37th and O Sts. NW, Washington, DC 20057 (Received 18 December 2019; accepted 21 July 2020) We present a full algebraic derivation of the wavefunctions of a simple harmonic oscillator. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation of the coordinate-space or momentum-space wavefunctions from the energy eigenvectors. It is simple to incorporate into the undergraduate and graduate curricula. We provide a summary of the history of operator-based methods as they are applied to the simple harmonic oscillator. We present the derivation of the energy eigenvectors along the lines of the standard approach that was first presented by Dirac in 1947 (and is modified slightly here in the spirit of the Schrodinger€ factorization method). We supplement it by employing the appropriate translation operator to determine the coordinate-space and momentum-space wavefunctions algebraically, without any derivatives. VC 2020 American Association of Physics Teachers. https://doi.org/10.1119/10.0001702
I. INTRODUCTION from the geometrical fact that it is equal to the product of the length of each vector multiplied by the cosine of the angle The Hamiltonian of the simple harmonic oscillator is between them. This suggests that a “representation- independent” derivation of wavefunctions might be possible; p^2 1 H¼^ þ mx2x^2; (1) indeed, this is achieved by using the translation operator to 2m 2 0 represent the position eigenvector in terms of the position eigenvector at the origin and additional operator manipula- where p^ and x^ denote the momentum and position operators, tions. We provide details below for the simple harmonic which satisfy the canonical commutation relation oscillator. We believe that this should become part of the ½ x^; p^ ¼ x^p^ p^x^ ¼ ih (2) standard treatment of the simple harmonic oscillator. We do want to point out that both Bohm€ 1 and (the third edition of) 2 (hats will be used on all operators throughout this work). Merzbacher also showed how to compute wavefunctions in a representation-independent fashion, but their approach Here, we have the mass m and the frequency x0 of the oscil- lator. Most textbooks solve this problem in two ways: (1) develops recurrence relations between the wavefunctions of first, one represents the momentum operator in coordinate different energy eigenstates at the same position (and hence space via p^ ¼ ih ðd=dxÞ and solves the resulting differential is different from our approach). The key to our procedure equation, finding the energy eigenvalues via the condition lies in employing the appropriate translation operators to that the solution be bounded as jxj!1and (2) an abstract relate the components of a wavefunction to each other (same operator method is employed to factorize the Hamiltonian eigenstate, different position); this then allows for the entire and is then used to determine the energy eigenvalues and a wavefunction to be determined algebraically from its value representation-independent form of the eigenvectors. When at one (spatial) point (which is ultimately determined by nor- it comes time to determine the wavefunctions in the latter malization). While we do not elaborate further on this point case, one converts the lowering operator into the coordinate- here, this methodology employing translation operators can space representation, which yields a first-order differential be used to find the wavefunctions of many other quantum- equation for the ground state. Then applying the raising mechanical potentials. Examples for particles in square-well operators in the coordinate representation to the ground state potentials can be found in Ref. 3. produces the excited state wavefunctions in coordinate Before jumping into the derivation, we briefly summarize space; a similar approach can also be used in momentum the Schrodinger€ factorization method for determining the space. We want to clarify one way to interpret what a wave- energy eigenvalues and eigenstates of the simple harmonic 4 5 function is. In the coordinate representation, the basis vectors oscillator following the textbooks of Green and Ohanian are the eigenvectors of position given by jxi, which satisfy because the method is not well known to many 6 x^jxi¼xjxi. These eigenvectors are known to produce an (Schrodinger’s€ original reference is also quite readable ). orthonormal basis set by the spectral theorem for essentially We do so here to present the context for our slight change in self-adjoint operators. A coordinate-space wavefunction is the standard algebraic derivation of the simple harmonic constructed by calculating the components of a quantum oscillator eigenstates. We employ the Dirac notation for state vector jwi along all of the basis vectors of the coordi- states in the Hilbert space throughout this work. nate representation, and can be thought of as the set While Schrodinger’s€ discovery of the Schrodinger€ equa- fhxjwi : for all xg. What is interesting about this observation tion is widely known today, his work from the 1940s on the is that each component of the coordinate-space wavefunc- so-called factorization method is less familiar. This portion tion, i.e., each element of the set fhxjwi : for all xg,isan of Schrodinger’s€ work has been omitted from most quantum inner product of just one position eigenvector with the quan- textbooks with the exception of its application to the har- tum state vector. The inner product between any bra and any monic oscillator, the simplest example of this technique. The ket is just a property of the bra and the ket as is well known general factorization method may appear rather abstract, but
976 Am. J. Phys. 88 (11), November 2020 http://aapt.org/ajp VC 2020 American Association of Physics Teachers 976 it can be straightforwardly applied to an array of problems. for any state vector jwi. Hence, we learn that the ground In fact, any problem that can be solved via Schrodinger’s€ dif- state j0i of the simple harmonic oscillator requires ferential equation can also be solved using the factorization method. Details can be found in the above references. a^j0i¼0; (9) The goal of the factorization method is to factorize the Hamiltonian in the form and the ground-state energy is E0 ¼ h x0=2. We next find the relevant intertwining relationship: we † † H¼^ A^ A^ þ E (3) operate a^ on the right side of Eq. (7) and discover that † † 1 † † † 1 and one can immediately verify that H^ a^ ¼ h x0 a^ a^ þ a^ ¼ h x0a^ a^a^ þ ÂÃ2 2 1 † 1 † ^ A^ ¼ pffiffiffiffiffiffi ðp^ imx0x^Þ and A^ ¼ pffiffiffiffiffiffi ðp^ þ imx0x^Þ ¼ a^ Hþh x0 ; (10) 2m 2m (4) where the last line follows by applying the commutation relation of the Dirac operators. We then immediately find achieves this factorization for the simple harmonic oscillator that the eigenstates satisfy with E ¼ h x=2. We chose this nonstandard notation because it matches the notation for the ladder operator method of the ðÞ† n a^ ffiffiffiffi simple harmonic oscillator given in many early quantum text- jni¼ p j0i; (11) n! books. However, the method and notation for the algebraic solution to the harmonic oscillator differ somewhat in today’s with energies texts. The abstract method was first introduced in the 1930 edition of Dirac’s textbook on quantum mechanics7 (first edi- 8 1 tion) and further developed in his 1947 edition (third edition); En ¼ h x0 n þ : (12) 2 a more complete history is developed below. The framework for the operator method has remained unchanged, but a differ- This derivation repeatedly uses the intertwining relation to ent notation has since been universally adopted by quantum determine the energy and the normalization. Finally, we textbooks. The i factors are moved from the coordinate to the assume the ground state j0i is normalized from the beginning ^ ^† momentum, and we work with dimensionless a and a rather (h0j0i¼1). This derivation differs from the standard € than the Schrodinger operators. The dimensionless (Dirac) approach, but we think it works better logically since it first ladder operators are then defined as determines the ground state from the factorization and a pos- rffiffiffiffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiitivity argument and then constructs the excited states mx p^ mx p^ a^† ¼ 0 x^ i ; a^ ¼ 0 x^ þ i : directly from the intertwining relation. Normalization then 2h mx0 2h mx0 follows as the last step. (5) Before developing the algebraic derivation of the wave- pffiffiffiffiffiffiffiffi function, we describe the historical background for the sim- These operators differ by a factor of 6i= h x0 from the cor- ple harmonic oscillator. responding Schrodinger€ operators given in Eq. (4). We work now with the modern Dirac form of these operators due to II. HISTORY OF THE SIMPLE HARMONIC their familiarity. OSCILLATOR IN QUANTUM MECHANICS Our next task is to establish the eigenvectors and eigenval- ues of the simple harmonic oscillator following the Although much work has been done on the history of Schrodinger€ approach. This methodology is different from quantum mechanics, it seems no one has attempted an in- Dirac’s 1947 approach, which relies too heavily on the depth exploration of the harmonic oscillator. There is no matrix mechanics approach in that it exploits the raising and mention in standard quantum historical texts, including lowering operators to move up and down the spectrum. It is Jammer,10 Taketani and Nagasaki’s11 three-volume work, more closely aligned with the approach of Ikenberry,9 which and even Mehra and Rechenberg’s12 six-volume set on the employs instead the 1940 Schrodinger€ notion of positivity as history of quantum mechanics. In his discussion of transfor- the critical criterion for determining eigenstates after facto- mation theory, Purrington13 does mention the introduction of rizing a Hamiltonian. Here is how it is done. ladder operators for the harmonic oscillator in Born and The (Dirac) raising and lowering operators satisfy Jordan’s textbook.14 However, our interpretation of Born and Jordan’s book differs from that of Purrington, as we read ÂÃmx i a^; a^† ¼ 0 2½ ¼p^; x^ 1(6)the Born and Jordan text as working with Heisenberg matri- 2h mx0 ces of the raising and lowering operators. Thus, we do not consider their approach an abstract operator formalism. and While the aforementioned texts expound on the evolution of a variety of areas in quantum mechanics, none of them trace † 1 the progression of the solutions of the harmonic oscillator. H¼^ h x0 a^ a^ þ : (7) 2 One explanation for this might be a simple lack of interest in the harmonic oscillator during the early development of † Since a^ a^ is a positive semidefinite operator, it satisfies quantum theory. Most of the original publications that devel- oped quantum mechanics in the period from 1925-30 were † 2 hwja^ a^jwi¼ka^jwik 0(8)primarily interested in determining the atomic spectra of
977 Am. J. Phys., Vol. 88, No. 11, November 2020 M. Rushka and J. K. Freericks 977 elements other than hydrogen and in quantizing light. In the wavefunction but also developed the differential equation addition, the simple harmonic oscillator spectrum was deter- method for treating the harmonic oscillator. Using his time- mined in the first matrix mechanics papers by Heisenberg15 independent wave equation for a harmonic potential and Born and Jordan.16 Schrodinger€ solved it in his second d2w ðqÞ 2m 1 paper,17 providing both the spectrum and the wavefunctions n 2 2 2 þ 2 En mx0q wnðqÞ¼0; (15) (via a differential equations approach). So the harmonic dq h 2 oscillator seems to have slipped through the cracks, and its historical study remains underdeveloped. Starting from the Schrodinger€ found the energies of the harmonic oscillator as 1920s, we seek here to provide an understanding of well as its eigenstates, which he expressed (unnormalized) in the development of the quantum-mechanical solutions of the the coordinate-space representation as simple harmonic oscillator. Note that from time to time we rffiffiffiffiffiffi! 2 2 x0 will use the original notation employed in the original w ðqÞ¼eð mx0=2h Þq H q ; (16) articles. We try to make it clear when this is being done n n h below.
Heisenberg was the first to find the energies of the har- where Hn denotes the Hermite polynomials. Schrodinger€ monic oscillator in his 1925 paper15 that invented modern thus introduced the differential equation method now univer- quantum mechanics. His seminal paper relied on classical sally employed in all quantum textbooks, and his articulation equations of motion and replaced them with their matrix- of the eigenstate enabled the development of the operator valued quantum counterparts (a strategy similar to the old method in early editions of Dirac’s textbook.7,8 Dirac, like quantum mechanics method of Bohr-Sommerfeld quantiza- his contemporaries, discussed matrix mechanics in his 1930 tion). Using this matrix-valued equation of motion and the textbook. Indeed, the relationship between matrix mechanics canonical commutation relation, Heisenberg was able to find and operator methods is quite close. the quantized energy levels. The first problem treated was Before jumping into the development of the ladder opera- that of an anharmonic oscillator with a third-order perturba- tor method for the harmonic oscillator, we must mention the tion term. Heisenberg truncated his result to determine the appearance of bosonic creation and annihilation operators in energies for the unperturbed harmonic oscillator other areas of quantum theory. As noted earlier, a principal concern of many early quantum papers was the quantization 1 of light. Consequently, Dirac,21 Jordan,22 and Fock23 all pub- W ¼ h x n þ : (13) 0 2 lished papers in the late 1920s and early 1930s which include bosonic creation and annihilation operators. While at the While Heisenberg’s article provided essentially no details time it appears that they were unaware of the relation for how the calculation was done,18 he did compute the cor- between these operators and the harmonic oscillator, their rect result. Born and Jordan published a paper16 shortly after publications coincide with the origins of the ladder operator Heisenberg’s in which they provided the details of the method presented here. Since it was present in other areas of matrix-mechanics solution for the simple harmonic oscilla- quantum theory at the time, we can see then that the notion tor. The matrix mechanics methodology does contain many of ladder operators was not unique to the early treatment of elements of the operator method which Dirac later developed the harmonic oscillator. in the first three editions of his textbook.7,8,19 Matrix We also mention one other item which was of great inter- mechanics works by essentially determining the properties est to the quantum pioneers—the theory of canonical trans- of the position space matrix, defined in modern terms via formations and the formulation of quantum mechanics in terms of action-angle variables. Here, Dirac led the way in ði=h ÞHt ð i=h ÞHt his first quantum paper24 on canonical quantization, where qmnðtÞ¼hmje qe^ jni: (14) he nearly constructed the raising and lowering operators One can see that the time-dependence of the matrix goes like toward the end of the paper. He did note that the approach works for the simple harmonic oscillator but provided no exp ½ iðEn EmÞt=h . Substituting into the classical equa- tion of motion for the simple harmonic oscillator yields the details. Fritz London produced similar work in a 1926 paper,25 although the raising and lowering operators do not constraint that En Em ¼ 6h x0. Hence, the q^ matrix is tri- diagonal, and the consecutive energy levels are separated in explicitly appear in his work either. The first work to formally define two operators which factor- steps of h x0. Next, the positivity of the Hamiltonian is used ize the Hamiltonian of the harmonic oscillator is Born and to show that there must exist some minimum energy level 14 1 Jordan’s 1930 textbook, which was completed a few months equal to 2 h x0. From this ladder of energies, they deduced 7 that the nth diagonal value of the Hamiltonian is given by before Dirac’s first edition. They write the Hamiltonian as Heisenberg’s result in Eq. (13). The connection between 1 a Born and Jordan’s paper and the ladder operator method is H¼ p2 þ q2; (17) further exhibited in Birtwistle’s textbook,20 which presents 2l 2 diagrams in a ladder formation connecting the different where l represents mass and a what they call the quasi- energy levels. elastic constant. Born and Jordan introduced two matrices These matrix-mechanics papers failed to treat the eigen- † states of the harmonic oscillator since matrix mechanics has b ¼ Cðp 2pi 0lqÞ and b ¼ Cðp þ 2pi 0lqÞ; € no concept of an eigenfunction. It was not until Schrodinger (18) introduced the wavefunction in 1926 that quantum papers ffiffiffiffiffiffiffiffiffiffiffiffi p † † began to explicitly refer to the eigenstates of the harmonic where C ¼ 1= 2h 0l. They noted that bb b b ¼ 1 and oscillator. In his paper,17 Schrodinger€ not only introduced rewrote the Hamiltonian as
978 Am. J. Phys., Vol. 88, No. 11, November 2020 M. Rushka and J. K. Freericks 978 ix † h 0 † h 0 which he denoted via the unconventional notation e and H¼h bb ¼ h b b þ : (19) ix 0 2 0 2 e , respectively. He noted that we can write the momen- tum and position operators as Born and Jordan’s definition of b and b†, and subsequent rffiffiffiffiffiffiffi rewriting of the Hamiltonian appears nearly identical to the mx 1=2 ix ix 1=2 † p^ ¼ J^ e þ e J^ and modern operator method (which instead uses a^ and a^ ). 2 Although they referred to them as “Stufenmatrizen,” Born rffiffiffiffiffiffiffiffiffiffi † 1 1=2 1=2 and Jordan did not seem to use b and b as ladder operators, q^ ¼ iJ^ eix þ ie ixJ^ ; (25) which act directly on eigenstates. We then do not consider 2mx this approach to be the initial formulation of the abstract ^ operator method. Born and Jordan apparently wrote their where J was denoted the “action variable” and given by 1930 textbook as a last-ditch-effort to save matrix mechanics H^ 1 from oblivion. This did not happen, and unfortunately the J^ ¼ hI: (26) textbook has been nearly forgotten (in part because it was x 2 never translated into English). The operator method for the simple harmonic oscillator With Eqs. (24) and (26), we can calculate that then takes its first form in the 1930 edition7 of Dirac’s text- pffiffiffi pffiffiffi 1=2 ix † ix 1=2 book, although his discussion was quite similar to Born and J^ e ¼ ha^ and e J^ ¼ h a^; (27) Jordan’s and inherits much of the matrix-mechanics argu- † ment. Dirac worked with a dimensionless abstract where a^ and a^ are the ladder operators commonly used to Hamiltonian first. To find the eigenvalues of treat the harmonic oscillator today (but not introduced by Dirac in 1930). Furthermore, with Eq. (27) above, we can H¼^ p^2 þ q^2; (20) also see that the form of Eq. (25) is almost identical to the way the momentum and position operators are defined today Dirac defined an operator A^ as follows (note A^ is not a ladder in terms of the ladder operators. Finally, Dirac’s 1930 text- operator here): book seems to be the first to give the wavefunctions of the harmonic oscillator as the overlap of the energy eigenstates A^ ¼ðp^ þ iq^Þðp^ iq^Þ: (21) with position space, which he wrote as an inner product ðqjnÞ, where jnÞ denoted the nth eigenstate (this was written A simple calculation showed A^ to be essentially the before Dirac notation was introduced). Dirac used differen- Hamiltonian for the harmonic oscillator. He defined the tial equations to find the wavefunctions, which he expressed eigenstates of A^ to satisfy the standard eigenvalue equation with a finite power series in q (using the standard Frobenius series solution method). While the operator method in his A^jA0i¼A0jA0i (22) 1930 textbook contains remarkable similarities to that in modern textbooks, there remain a few differences to point and then proceeded through a matrix-mechanics argument to out. Dirac did not formally define the ladder operators here show that hA0jðp^ þ iq^ÞjA00i equals zero unless A00 ¼ A0 2. but instead used expressions of the form ðp^6iq^Þ as ladder Using this and the non-negativity of p^2 þ q^2, Dirac found operators—indeed, his approach presaged the Schrodinger€ that the eigenvalues of A^ are all the even non-negative inte- factorization method since it is focused on factorizing the gers: 0; 2; 4; 6; … and so on. From his earlier assertion that Hamiltonian. We also note that Dirac included the factor of i on the position operator in Eqs. (21) and (25) above, which 0 00 hA jðp^ þ iq^ÞjA i¼dA00;A0 2; (23) differs from the standard notation today, but again agrees with the Schrodinger€ factorization method. However, it is we can then see how ðp^ þ iq^Þ acts as a ladder operator on also fair to say that Dirac’s approach is quite similar to the jA00i to raise it to the next highest eigenstate of A^.Dirac’s matrix mechanics methodology of Born and Jordan. Dirac expression given in Eq. (21) then showed that A^ is analogous used the Heisenberg matrices to determine the eigenvalues to the ladder operator formulation of the Hamiltonian. What in a standard matrix mechanics approach. His main differ- Dirac’s initial treatment lacked was a formulation of the ence is that he was the first to work with the operators by eigenstate in terms of operators acting on the ground state themselves instead of solely with the matrices (which is how (which we conjecture is because he adopted a matrix- we interpret the Born and Jordan methodology). mechanics methodology to find the spectrum and matrix- Other textbooks in the 1920s and 1930s do not treat the mechanics does not construct eigenstates). Dirac alluded to simple harmonic oscillator by operator methods but usually the ladder operators by introducing their matrix representation do so by both matrix mechanics and by wave mechanics. 0 1 0 1 This includes texts like Birtwistle (1928),20 Condon and 00000 01000 Morse (1929),26 Born and Jordan (1930),14 Mott (1930),27 B C B C 28 29 30 B C B C Sommerfeld (1930), Fock (1932), Frenkel (1932), B10000 C B00100 C 31 32 B C B C Pauli (1933), Frenkel (1934), Pauling and Wilson B01000 C B00010 C 33 34 35 B C B C (1935), Jordan (1936), Kemble (1937), and Dushman B C and B C; 36 B00100 C B00001 C (1938). The one exception from the 1930s appears to be B C B C Rojansky’s 1938 text,37 which provides a treatment nearly B C B C @00010 A @00000 A identical to Dirac’s 1930 method. But Rojansky makes it ...... clear that he is working with operators (as his derivation is in ...... a chapter entitled “The Symbolic Method”), and he strictly (24) works solely with the operators, never introducing the
979 Am. J. Phys., Vol. 88, No. 11, November 2020 M. Rushka and J. K. Freericks 979 Heisenberg matrices in this section of his book (although he He then gives the real and imaginary parts of As by does discuss matrix mechanics elsewhere). While he has all ffiffiffi of the elements available to construct the eigenvector 1 p ReAs ¼ ðAs þ A Þ¼ pQs abstractly in terms of the raising operators, he fails to do so. 2 s He does, however, employ the intertwining relationship in 1 pffiffiffi and ImA ¼ ðA A Þ¼ pP ; (32) the derivation, making it closer to the way we proceeded s 2i s s s here. Intriguingly, Schrodinger€ 6 developed his factorization which bears a striking resemblance to the way many popular method in 1940–1941. The first problem he tackled was the textbooks relate p^ and q^ to the ladder operators. If our suspi- simple harmonic oscillator. In this work, he showed that one cions hold true, the use of a would then stand for can evaluate the equation a^j0i¼0 for the ground state (in “amplitude.” The origin of this notation would then lie in the coordinate space) and found a first-order differential equa- early work on quantizing light by the fathers of modern tion for the ground-state wavefunction. He then simply stated quantum mechanics. Born and Jordan’s textbook14 also that one can extend the same method to higher eigenstates seems to support this notion, as they explicitly referred to b but provided no details. Hence, Schrodinger€ was, perhaps and b† as “komplexe Amplituden.” One should also note that aptly, the first to determine all the eigenvectors (and the Frenkel’s 1934 book32 discussed many of these same themes associated wavefunctions) for the simple harmonic oscillator too when quantizing light, including the same modern nota- via the operator-based approach. tion as used by Schiff fifteen years later. Frenkel’s approach The next development of the operator method for the sim- was deeply entrenched in matrix mechanics, as was much of ple harmonic oscillator appears in the 1947 edition of the work at that time—our interpretation is that the objects Dirac’s textbook.8 This gives the origin of the modern he worked with were in fact matrices and not abstract opera- approach adopted by all subsequent textbooks and provides tors in their full generality—but this conclusion is not crystal the modern abstract derivation. Dirac explicitly defines clear. Interestingly, Frenkel also employed the a; a† notation dimensionless operators when quantizing light. rffiffiffiffiffiffiffiffiffiffiffiffi Through the 1950s and 1960s, we see textbooks use a 1 g ¼ ðp^ þ imxq^Þ combination of differential equations and Dirac’s 1947 oper- 2mh rxffiffiffiffiffiffiffiffiffiffiffiffi ator method to treat the harmonic oscillator. While every 1 book’s operator treatment follows Dirac’s, we see a swathe and g ¼ ðp^ imxq^Þ; (28) 40 2mh x of different notations. We find this in Bohm, Landau and Lifshitz,41 Messiah,42 Dicke and Wittke,43 Merzbacher,2 44 45 46 which he uses to establish this modern operator method. He Powell and Crasemann, Harris and Loeb, Park, 47 4 48 49 checks that Gottfried, Green, Ziman, and Flugge.€ From the late 1960s to date, all quantum textbooks use the same notation gg gg ¼ 1 (29) as Schiff, Messiah, and Park. These include Saxon,50 Baym,51 Gasiorowicz,52 Cohen-Tannoudji,53 and Winter54 in and shows that g and g act as ladder operators which raise addition to virtually all subsequent textbooks. We could not and lower the energy of the harmonic oscillator in steps of figure out why all textbooks adopted a standardized notation h x, respectively. Dirac demonstrates that gg is a positive after 1970, but the earliest instance of the modern approach semi-definite operator and uses this to show that the ground with the modern notation seems to be in Messiah’s 1959 1 42 state energy of the harmonic oscillator equals 2 h x.He textbook. expresses the nth energy eigenstate as gnj0i and represents In summary, we see the operator method for the simple the wavefunctions by harmonic oscillator to have developed as follows. The matrix 15 16 0 n mechanics approach of Heisenberg and Born and Jordan hq jg j0i; (30) already has about one third of the abstract method worked out. That approach uses the positivity of the Hamiltonian and which he finds using differential equations. While Dirac’s a ladder structure of the matrix elements to determine the method here is identical to the modern operator method used energy eigenvalues. The ladder operation structure was even today, his notation differs slightly. He uses g and g to denote illustrated graphically by Birtwistle.20 Next, Born and the ladder operators and again includes the factor of i on the Jordan’s 1930 textbook14 was the first to represent the ladder position operator. It is fair to say that it is here, in 1947, that operators in the matrix mechanics formalism, but Dirac’s today’s popular abstract formulation of the simple harmonic 1930 textbook7 initiated the abstract operator approach with oscillator is born. the factorization of the Hamiltonian in terms of operators, The remainder of the harmonic oscillator’s development even though it later employed the matrix mechanics method- consists mainly of notational changes. Leonard Schiff intro- ology to determine the eigenvalues. Rojansky37 performed duced, but did not significantly use, the a^ and a^† notation in the first completely abstract derivation free from matrix his 1949 quantum textbook.38 We suspect the reason for the mechanics. Though he was on the precipice of also determin- use of this letter to denote the ladder operators may lie in the ing the eigenvectors, he did not. That had to wait for Fock second volume of Sin-Itiro Tomonaga’s 1953 quantum text- 23 39 space and Schrodinger’s€ use of it in his factorization book. Tomonaga uses As to denote the complex time- 6 dependent amplitude of a De Broglie wave packet method before one could construct the eigenvectors abstractly (but the derivation still required going to coordi- X1 nate space to determine the wavefunctions). Finally, Dirac 8 Wðx; y; z; tÞ¼ AsðtÞ/ðx; y; zÞ: (31) finished the modern derivation in his 1947 text. The opera- s¼1 tor method was immediately adopted by nearly all other
980 Am. J. Phys., Vol. 88, No. 11, November 2020 M. Rushka and J. K. Freericks 980 textbooks, although the notation did not become the standard This relation is often called the braiding relation. When one we are accustomed to until the early 1970s. ½A^; B^ commutes with A^ and B^, we then have the exponential re-ordering identity III. ALGEBRAIC DERIVATION OF THE A^ B^ B^ A^ ½ A^;B^ WAVEFUNCTIONS OF THE SIMPLE HARMONIC e e ¼ e e e ; (36) OSCILLATOR which includes a correction term when the exponential oper- We begin the algebraic derivation of the wavefunctions by ators are re-ordered. simply noting that the components are the inner products of To start working with the translation operator, we use the the energy eigenvectors jni with the position jxi and momen- Hadamard lemma in Eq. (33), which allows us to evaluate tum jpi eigenvectors, or wnðxÞ¼hxjni and /nðpÞ¼hpjni. the similarity transformation of the operator x^ as follows Our strategy is to employ operator methods without resorting (with x0 being a real number): to specific representations of the operators, so we do not i x2 need to introduce the coordinate-space representation of the ði=h Þx0p^ ð i=h Þx0p^ 0 e xe^ ¼ x^ þ x0½ p^; x^ ½ p^; ½ p^; x^ þ momentum operator in terms of a derivative with respect to h 2h 2 the position. Instead, we follow the representation- ¼ x^ þ x : (37) independent operator-based approach initiated by Pauli55 0 56 and independently by Dirac in 1926. The final equality occurs because ½p^; x^ ¼ ih is a number, We assume that an eigenstate exists for position at the ori- not an operator, and subsequently it commutes with all addi- gin and is denoted jx ¼ 0i. It satisfies x^jx ¼ 0i¼0, and we tional multiple commutators of p^. This truncates the relate the component hx ¼ 0jni to all other components of Hadamard lemma expression after the first commutator. the coordinate-space wavefunction. Note that we do not need Next, we multiply both sides of Eq. (37) by exp ð ix0p^=h Þ to worry about the normalization of the state for anything from the left to yield that we do here, so we do not discuss this issue further (as its treatment is well covered in all quantum texts). ð i=h Þx0p^ ð i=h Þx0p^ xe^ ¼ e ðx^ þ x0Þ: (38) We will employ the Hadamard lemma, which is given by
With this identity, we establish the eigenvector jx0i, which X1 h ÂÃÂÃ ^ ^ 1 satisfies x^jx i¼x jx i (here, x is a number and a label for eA Be^ A ¼ B^ þ A^; A^; …; A^; B^ ; (33) 0 0 0 0 m! m the Dirac ket) m¼1 jx i¼eð i=h Þx0p^jx ¼ 0i: (39) where the m subscript on the commutators denotes that there 0 are m nested commutators; this lemma is also called the Operating x^ onto the state jx0i yields Baker-Hausdorff lemma and the braiding relation. But as far as we can tell, it was first discovered by Campbell in 1897 ð i=h Þx0p^ ð i=h Þx0p^ x^jx0i¼xe^ jx ¼ 0i¼e ðx^ þ x0Þjx ¼ 0i [see Eq. (19) of the historical discussion of the Baker- Campbell-Hausdorff relation57] and hence should be called ¼ x0jx0i: (40) the Campbell lemma. Despite significant research, we were unable to determine where the Hadamard lemma name The last equality follows from x^jx ¼ 0i¼0, the fact that comes from. numbers always commute with operators and the definition Before we jump into the derivation of position and of jx0i. Hence, Eqs. (39) and (40) establish that jx0i is an momentum operators, we note that the Hadamard lemma can eigenstate of x^ with eigenvalue x0. be employed to establish some additional identities. Any Similarly, one can also derive that the momentum eigen- function f ðB^Þ of an operator B^ that can be written as a power states satisfy ^ series in B satisfies ði= hÞp0x^ jp0i¼e jp ¼ 0i; (41) ÀÁ X1 X1 m A^ A^ A^ m A^ A^ A^ e f ðB^Þe ¼ e fmB^ e ¼ fm e Be^ where p0 is both a number and the label for the ket. Note the m¼0 m¼0 different sign in the exponent of the operator for the position and momentum eigenvectors. A^ ^ A^ ¼ f ðe Be Þ We are almost ready to compute the coordinate-space X1 1 Â Â ÂÃÃ wavefunction using purely algebraic methods. The deriva- ¼ f B^ þ A^; A^; …; A^; B^ : tion requires one more identity: the Baker-Campbell- ! m 58–60 m¼1 m Hausdorff (BCH) identity. The BCH identity is (34) “halfway” between the two sides of the exponential re- ordering identity, which rewrites the exponential of the sum This is an exact relation. Choosing f ðB^Þ¼exp ðB^Þ then of the operators in terms of the two exponential operators yields an important identity after some simple re-arranging and a correction factor—here, the BCH formula takes a of terms: product of exponential of operators and rewrites it as the exponential of a new operator. Unlike the Hadamard lemma X1 Â Â ÂÃÃ ^ ^ 1 ^ and its application to exponential re-ordering, the BCH iden- eA eB ¼ exp B^ þ A^; A^; …; A^; B^ eA : m! m tity does not have any simple explicit formula for its result in m¼1 the general case (although one can write the result in closed (35) form).61,62 Fortunately for us, we need it only for the case
981 Am. J. Phys., Vol. 88, No. 11, November 2020 M. Rushka and J. K. Freericks 981 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ½A^; B^ commutes with A^ and B^—in this case, the BCH 1 2 † w ðxÞ¼pffiffiffiffi e ðmx0=4h Þx hx ¼ 0je ðmx0=2h Þxa^ result greatly simplifies and is given by n n ! ! rffiffiffiffiffiffiffiffiffi n mx A^ B^ A^þB^þ1 A^;B^ B^ A^ A^þB^ 1 A^;B^ † 0 e e e 2½ and e e e 2½ : (42) a^ þ x jn ¼ 0i: (48) ¼ ¼ 2h
The BCH identity is a well-known and well-established Next, we introduce a new exponential factor with the oppo- result, so we do not provide its derivation here; in this form, site sign of the exponent multiplying the ground-state wave- it is often called the Weyl identity. function, because it equals 1 when operating against the We now have all the technical tools needed to determine state: the coordinate-space wavefunction w ðxÞ¼hxjni. Using the n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi position eigenstates and the energy eigenstates, we immedi- 1 2 † w ðxÞ¼pffiffiffiffi e ðmx0=4h Þx hx ¼ 0je ðmx0=2h Þxa^ ately find that n n! rffiffiffiffiffiffiffiffiffi ! n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ði=h Þxp^ † n mx0 w ðxÞ¼hxjni¼pffiffiffiffi hx ¼ 0je ðÞa^ jn ¼ 0i: (43) a^† þ x e ðmx0=2h Þxa^jn ¼ 0i: n n! 2h
The operators p^ and a^† can be easily identified by their hats. (49) Note that one can think of this representation in the follow- The general functional braiding relation is used again to ing way: at the origin, the wavefunction is w ð0Þ¼hx ¼ n bring the rightmost exponential factor to the left through the 0jni (which is a number that will ultimately be fixed by nor- a^† term raised to the nth power malization) and the translation operator then shifts the wave- pffiffiffiffiffiffiffiffiffiffiffiffiffiffi function from the origin to the position x and tells us how the 1 2 † ffiffiffiffi ðmx0=4h Þx ðmx0=2h Þxa^ wavefunction value changes in the process. This allows us to wnðxÞ¼p e hx ¼ 0je n! ! compute the wavefunction everywhere by shifting the value rffiffiffiffiffiffiffiffiffiffiffiffi n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi of the coordinate. The algebraic computation then simply 2mx0 e ðmx0=2h Þxa^ a^† þ x jn ¼ 0i: evaluates the operator expression. h The strategy to determine the wavefunction algebraically now takes a few additional steps. First, we replace the (50) momentum operator in the exponent of the translation opera- tor by its expression in terms of the ladder operators Now, we use the BCH relation again to combine the two rffiffiffiffiffiffiffiffiffiffiffiffi exponentials into one which increases the Gaussian exponent mh x by a factor of two, 0ðÞ† p^ ¼ i a^ a^ : (44) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 2 † ffiffiffiffi ðmx0=2h Þx ðmx0=2h Þxða^ þa^Þ wnðxÞ¼p e hx ¼ 0je The wavefunction becomes n ! ! rffiffiffiffiffiffiffiffiffiffiffiffi n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi † 2mx0 1 ^ ^† † n a^ þ x jn ¼ 0i: (51) w ðxÞ¼pffiffiffiffi hx ¼ 0je ðmx0=2h Þxða a ÞðÞa^ jn ¼ 0i: (45) h n n!
Then we use the first BCH relation in Eq. (42) with A^ / a^† Finally, we use the fact that the sum of the raising and lower- and B^ / a^ to factorize the translation operator into a factor ing operator is proportional to the position operator involving the raising operator on the left and the lowering rffiffiffiffiffiffiffiffiffiffiffiffi operator on the right. This is given by h x^ ¼ ðÞa^ þ a^† : (52) pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2mx0 1 2 † ffiffiffiffi ðmx0=4Þhx ðmx0=2h Þxa^ wnðxÞ¼p e hx ¼ 0je n! ffiffiffiffiffiffiffiffiffiffiffiffiffiffi We replace the sum of the raising and lowering operator in p n e ðmx0=2h Þxa^ðÞa^† jn ¼ 0i: (46) the exponent and let it act on the state to the left, where it gives 1, because the position operator annihilates the state Third, we take the relation in Eq. (34) and multiply by hx ¼ 0j. The wavefunction has now become ^ ! exp ðAÞ on the right to create the general functional braiding rffiffiffiffiffiffiffiffiffiffiffiffi n relation and apply it to the matrix element for the wavefunc- 1 2 2mx0 n ffiffiffiffi ðmx0=2 hÞx † ^ ^† wnðxÞ¼p e hx ¼ 0j a^ þ x jn ¼ 0i: tion with f ðBÞ¼ða Þ . This yields n! h pffiffiffiffiffiffiffiffiffiffiffiffiffiffi (53) 1 2 † w ðxÞ¼ pffiffiffiffi e ðmx0=4h Þx hx ¼ 0je ðmx0=2 hÞxa^ n n! rffiffiffiffiffiffiffiffiffi ! We are almost done. We have achieved a reduction of the n pffiffiffiffiffiffiffiffiffiffiffiffiffiffi mx0 problem into a Gaussian function multiplied by a matrix ele- a^† þ x e ðmx0=2h Þxa^jn ¼ 0i: (47) 2 h ment which is an nth degree polynomial in x. All that is left is evaluating the polynomial. To do this, we first introduce a definition of the polynomial, which we will then show is a The rightmost exponential factor gives 1 when it operates on so-called Hermite polynomial Hn. We write the wavefunc- the state because a^jn ¼ 0i¼0. Thus, we have tion as
982 Am. J. Phys., Vol. 88, No. 11, November 2020 M. Rushka and J. K. Freericks 982 rffiffiffiffiffiffiffiffiffi ! rffiffiffiffiffiffiffiffiffi !rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi ! 1 mx0 2 mx0 mx0 mx0 ffiffiffiffiffiffiffiffiffi ðmx0=2h Þx w ðxÞ¼p Hn x e hx ¼ 0jn ¼ 0i; Hn x ¼ 2 xHn 1 x n n!2n h h h h ! (54) rffiffiffiffiffiffiffiffiffi mx0 2ðn 1ÞHn 2 x : (59) which defines the Hermite polynomial via h ! rffiffiffiffiffiffiffiffiffi pffiffiffiffiffi n This recurrence relation, which is of the form HnðzÞ mx0 2 H x ¼ ¼ 2zHn 1ðzÞ 2ðn 1ÞHn 2ðzÞ, is the standard Hermite n h hx ¼ 0jn ¼ 0i !polynomial recurrence relation when H0ðzÞ¼1 and H1ðzÞ rffiffiffiffiffiffiffiffiffiffiffiffi n ¼ 2z, as we have here. 2mx hx ¼ 0j a^† þ 0x jn ¼ 0i: We have now established that the simple-harmonic-oscil- h lator wavefunction satisfies (55) rffiffiffiffiffiffiffiffiffi ! 1 mx 2 0 ðmx0=2h Þx w ðxÞ¼pffiffiffiffiffiffiffiffiffi Hn x e hx ¼ 0jn ¼ 0i: Note that the number hx ¼ 0jn ¼ 0i is the normalization con- n n!2n h stant for the ground-state wavefunction; we will discuss how to determine it below. This definition allows us to immedi- (60) ately determine the first two polynomials H and H . 0 1 The last task in front of us is to find the normalization factor. Choosing n ¼ 0 in Eq. (55) immediately yields H ¼ 1. 0 This is computed for the ground state via Choosing n ¼ 1, produces ð ! 1 rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi ffiffiffi 2 p 2 mx0=hx mx0 mx0 2 jhx ¼ 0jn ¼ 0ij dxe ¼ 1 (61) H x ¼ 2 x þ 1 1 h h hx ¼ 0jn ¼ 0i or hx ¼ 0ja^†jn ¼ 0i: (56) 1 mx 4 The second term vanishes for the following reason: we first hx ¼ 0jn ¼ 0i¼ 0 : (62) note that a^†jn ¼ 0i¼ða^† þ a^Þjn ¼ 0i, because the lowering ph operator annihilates the ground state. Hence a^†jn ¼ 0i / x^jn ¼ 0i. But hx ¼ 0jx^ ¼ 0, so this state vanishes when it We have finally produced the wavefunction for the simple acts against the position eigenstate. harmonic oscillator using algebraic methods. Note that cal- For the remainder of the Hermite polynomials, we work culus is only needed for the last normalization step. out a two-term recurrence relation. We focus on the nontriv- We end this section with a brief sketch of how one uses ial matrix element, and factorize the terms as follows: similar methods to determine the momentum-space wave- functions. To start, the momentum “translation” operator is ! ! rffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffi n 1 given by exp ðipx^=h Þ, and the momentum eigenstates satisfy † 2mx0 † 2mx0 hx ¼ 0j a^ þ x a^ þ x jn ¼ 0i: ði=h Þpx^ h h jpi¼e jp ¼ 0i: (63) (57) n The wavefunction is given by /nðpÞ¼ðiÞ hpjni; we added an additional global phase to ensure we reproduce the stan- The constant term in the first factor can be removed from dard results—you will see why this is important below. The the matrix element and it multiplies the matrix element with wavefunction can be expressed in terms of the operators as n – 1 operator factors (which is proportional to Hn 1). For † n the remaining term proportional to a^ , we replace the opera- ðiÞffiffiffiffi ð i=h Þpx^ † n † † † /nðpÞ¼p hp ¼ 0je ðÞa^ jn ¼ 0i: (64) tor by a^ ! a^ þ a^ a^. The term proportional to a^ þ a^ is n! proportional to x^, and so it annihilates when it operates on the left against the hx ¼ 0j state. The remaining a^ operator The remainder of the calculations proceeds as before for the can be replaced by the commutator of the n – 1 power of the coordinate-space wavefunction. We start by replacing the x^ a^† term, because a^jn ¼ 0i¼0. Generalizing the standard operator by the sum of raising and lowering operators; in this n n 1 result ½a^; ða^†Þ ¼nða^†Þ , the remaining commutator is case, the coefficients of the raising and lowering operators straightforward to evaluate via are now purely imaginary. We use BCH to factorize the 2 3 exponential into a raising operator on the left and lowering rffiffiffiffiffiffiffiffiffiffiffiffi ! n 1 operator on the right. Then we use the braiding identity to 4 2mx 5 a^; a^† þ 0x move the exponential through the ða^†Þn terms and let it oper- h ate on the ground state, where it produces 1. The shift term ! rffiffiffiffiffiffiffiffiffiffiffiffi n 2 added to the raising operator is now purely imaginary. Next, 2mx ¼ðn 1Þ a^† þ 0x : (58) we introduce a factor of 1 at the ground state, which is the h same exponential operator of the lowering operator but with the sign of the exponent changed. Then we use the braiding We can assemble all of these results to find the recurrence identity to bring it back to the left, BCH to place the opera- relation for the Hermite polynomials, which becomes tors in one exponential, and evaluate the momentum operator
983 Am. J. Phys., Vol. 88, No. 11, November 2020 M. Rushka and J. K. Freericks 983 on the momentum eigenstate. At this stage, the wavefunction operator to shift the wavefunction from the origin and com- has become pute the change of its value. It employs simple operator iden- tities (the Hadamard lemma and Baker-Campbell-Hausdorff n ðiÞ 2 identity when ½A^; B^ commutes with A^ and B^) and hence it is / ðpÞ¼pffiffiffiffi e p =2h x0m n easy to understand and follow even for undergraduates in an n! ! pffiffiffi n introductory course. In addition, we explored the history 2p hp ¼ 0j a^† i pffiffiffiffiffiffiffiffiffiffiffiffi jn ¼ 0i: (65) behind the operator method for the simple harmonic oscilla- h x0m tor. Our findings are that this history is much richer than sim- ply “Didn’t Dirac do that?” Indeed, we discovered that one- Notepffiffiffiffiffiffiffiffiffiffiffiffiffiffi the additionalffiffiffiffiffiffiffiffiffiffiffiffi factors of i and the replacement of third of the argument can already be found in the matrix p mechanics works of Heisenberg and Born and Jordan. We mx0= hx by p= h x0m. The Hermite polynomial now needs to be defined via argue that Dirac’s original 1930 treatment is much closer to the matrix mechanics approach and that it actually was pffiffiffiffiffi p 2nin Rojansky in 1938 who made the derivation a completely Hn pffiffiffiffiffiffiffiffiffiffiffiffi ¼ abstract operator argument. Even Schrodinger€ had a hand in h x m hp ¼ 0jn ¼ 0i 0 ! this, being the first to use the abstract operators to construct pffiffiffi n 2p eigenvectors and coordinate-space wavefunctions in 1940- hp ¼ 0j a^† i pffiffiffiffiffiffiffiffiffiffiffiffi jn ¼ 0i: 1941. Dirac then finished the methodology in 1947. h x0m We hope that our completion of this work here will be (66) adopted by others teaching quantum mechanics, as we feel it pffiffiffiffiffiffiffiffiffiffiffiffi is yet another beautiful demonstration of the elegance of the Starting with H0 ¼ 1 and H1 ¼ 2p= h x0m, we find the abstract operator approach. Now the entire simple harmonic same Hermitepffiffiffiffiffiffiffiffiffiffiffiffi polynomials as we found before, but now with oscillator problem can be solved algebraically! z ¼ p= h x0m. The rest of the calculation is similar to the coordinate space calculation. The normalization factor is ACKNOWLEDGMENTS found by a simple integral. One can see that this procedure will lead to the momentum-space wavefunction, which The initial stages of this work were supported by the finally satisfies National Science Foundation under Grant No. PHY-1620555 and the final stages under Grant No. PHY-1915130. In 1 1 p 2 addition, J.K.F. was supported by the McDevitt bequest at / p pffiffiffiffiffiffiffiffiffi H pffiffiffiffiffiffiffiffiffiffiffiffi e p =2h x0m: nð Þ¼ 1 n n Georgetown University. The authors thank W. N. Mathews 4 n!2 h x0m ðph x0mÞ Jr., for a critical reading of the manuscript and Manuel (67) Weber for the translation of the sections of the original 1930 German textbook of Born and Jordan cited below. The Aside from some different constants, the coordinate-space authors also thank Takami Tohyama for the help with and momentum-space wavefunctions have identical func- determining the original Japanese reference to Tomanaga’s tional forms. This is expected from the outset, because the textbook and Andrij Shvaika for the help with the reference Hamiltonian is quadratic in both momentum and position. to Fock’s original textbook. Hence, the wavefunctions must be isomorphic. This ends our algebraic derivation of the wavefunctions of a)Electronic mail: [email protected] the simple harmonic oscillator. Note that it used only the b)Electronic mail: [email protected] commutator ½x^; p^ ¼ih and the existence of eigenstates of 1Arno Bohm,€ Quantum Mechanics Foundations and Applications, 3rd ed. position at the origin and of the ground state of the simple (Springer-Verlag, New York, 1993). 2Eugen Merzbacher, Quantum Mechanics (Wiley, New York, 1961). harmonic oscillator. We hope that you will try employing it 3J. Alexander Jacoby, Maurice Curran, David R. Wolf, and James K. the next time you teach a quantum mechanics class. If you Freericks, “Proving the existence of bound states for attractive potentials do, we recommend having the students work out the in one and two dimensions without calculus,” Eur. J. Phys. 40, 045404 momentum-dependent wavefunctions as a homework prob- (2019). 4 lem after being shown the derivation of the coordinate-space Herbert S. 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P., New York, 1962). of the eigenvalues and eigenvectors to put them in a more 10Max Jammer, The Conceptual Development of Quantum Mechanics € (McGraw-Hill, New York, 1966). standard approach motivated by the Schrodinger factoriza- 11 tion methods instead of Dirac’s 1947 derivation. In addition, Masayuki Nagasaki and Mituo Taketani, The Formation and Logic of Quantum Mechanics (World Scientific, Singapore, 2001). we extended the operator-based method to also allow for an 12Jagdish Mehra and Helmut Rechenberg, The Historical Development of abstract derivation of the wavefunctions in coordinate and Quantum Theory (Springer-Verlag, Inc., New York, 1982). position space. This approach employed the translation 13Robert D. Purrington, The Heroic Age (Oxford U. P., New York, 2018).
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(Second Paper)*,” ibid. s1-29, 14–32 (1897). 36Saul Dushman, The Elements of Quantum Mechanics (Wiley, New York, 60F. Hausdorff, “Die symbolische exponentialformel in der gruppentheorie,” 1938). Ber. Verh. Saechs Akad. Wiss. (Leipzig) 58, 19–48 (1906). 37Vladimir Rojansky, Introductory Quantum Mechanics (Prentice-Hall, Inc., 61E. B. Dynkin, “Calculation of the coefficients in the Campbell-Hausdorff New York, 1938). formula,” Dokl. Akad. Nauk SSSR 57, 323–326 (1947) (in Russian). 38Leonard Schiff, Quantum Mechanics, 1st ed. (McGraw-Hill, New York, 62Ravinder R. Puri, Non-Relativistic Quantum Mechanics (Cambridge U. P., 1949). Cambridge, 2017); the derivation of the BCH formula is in Chap. 2.8.
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