Why are complex numbers needed in quantum mechanics? …

1 Why are complex numbers needed in quantum mechanics ?Some answers for the introductory levelRicardo Karam Department of Science Education, University of Copenhagen, DenmarkAbstractComplex numbers are broadly used in physics, normally as a calculation tool that makes thingseasier due to Euler s formula. In the end, it is only the real component that has physical meaning orthe two parts (real and imaginary) are treated separately as real quantities. However, the situationseems to be different in quantum mechanics , since the imaginary unitiappears explicitly in itsfundamental equations. From a learning perspective, this can create some challenges to this article, four conceptually different justifications for the use/need of complex numbers inquantum mechanics are presented and some pedagogical implications are INTRODUCTIONC omplex numbers were invented (or discovered?) in 16th-century Italy as a calculationtool to solve cubic equations.

2 In the beginning, not much attention was paid to theirmeaning, since the imaginary terms should cancel out during the calculations and only (real)roots were considered. Around 250 years later, complex numbers were given a geometricalinterpretation and, since then, they became quite a useful tool to reasonis that complex numbers represent direction algebraically (2D vectors) and many of theiroperations have a direct geometrical meaning ( , the product rule: multiply the normsand add the angles).It is particularly helpful to use complex numbers to model periodic phenomena, especiallyto operate with phase differences. Mathematically, one can treat a physical quantity as beingcomplex, but address physical meaning only to its real part. Another possibility is to treatthe real and imaginary parts of a complex number as two related (real) physical both cases, the structure of complex numbers is useful to make calculations more easily,but no physical meaning is actually attached to complex mechanics seems to use complex numbers in a more fundamental way.

3 It sufficesto look at some of the most basic equations, both in the matrix ([ p, x] = i~) and wave(i~ t= H ) formulations, to wonder about the presence of the imaginary unit. What isessentially different in quantum mechanics is that it deals with complex quantities ( wavefunctions and quantum state vectors) of a special kind, whichcannotbe split up into purereal and imaginary parts that can be treated independently. Furthermore, physical meaningis not attached directly to the complex quantities themselves, but to some other operationthat produces real numbers ( the square modulus of the wave function or of the innerproduct between state vectors).This complex nature of quantum mechanical quantities puzzled some of the very foundersof the theory. Schr odinger, for instance, was bothered by the fact that his wave functionwas complex and tried for quite some time to find physical interpretations for its (real) ,3It was probably not until Dirac formulated his bra-ket notation that it becameclearer that the complex quantities of quantum mechanics were of a different kind than theones commonly used in classical is therefore very reasonable to expect that introductory level students will face consid-2erable difficulties with the peculiar use of complex numbers in quantum mechanics .

4 However,it is not uncommon to find textbooks that postulate, from the get-go, that wave functionsand quantum state vectors are complex valued, without providing any sort of explanation forwhy this is the case. It is not only natural, but even desirable, that students inquire aboutthe reasons why certain mathematical structures are useful to describe physical , for newcomers to the field, one can say that it is perfectly legitimate to ask: Whydo we need complex numbers in quantum mechanics ? .There is certainly no unique straight answer to this question5 8; constructing plausiblearguments to address it is a matter of pedagogical creativity. The aim of this paper is topresent synthetic (and slightly modified) versions of four different justifications found in theliterature9and discuss some of their pedagogical implications. The first two justifications(IIA and IIB) align best with a position first approach, whereas the last two (IIC andIID) with a spin first approach.

5 The main selection criteria that guided this choice werethe plausibility of the arguments and their applicability for the very first lessons of anintroductory quantum mechanics WHY ARE complex numbers needed in quantum mechanics ?A. No information on position when momentum is known exactlyThis argument is found in R. Shankar s textbook10for the introductory level and goesas follows. Consider one particle in one dimension and assume de Broglie s matter wavesrelation ( =2 ~p), which is previously justified by empirical evidence from the double-slitexperiment with electrons. Since there appears to be a wave associated with the electron, itis reasonable to assume that there is a wave function (x) describing it (ignore time depen-dence). Postulate that the absolute square of this function (| (x)|2) gives the probabilitydensity, , the likelihood to find the particle between the positionsxandx+ last piece needed for the argument is inspired by Heisenberg s uncertainty is presented qualitatively in Shankar s textbook by thought experiments like the gamma-ray microscope.

6 One way to state the principle is to say that it is impossible to prepare aparticle in a state in which its momentum and position (along one axis) areexactlyknown. Now, suppose the electron is in a state of definite momentum. How would the wave3function for this state look like? According to de Broglie s relation, there is a wavelength as-sociated with the electron. Therefore, it seems plausible to assume a classical/real oscillatingwave function of the form p(x) =Acos2 x =Acospx~,(1)where p(x) denotes a state of definite momentum. p(x), as well as| p(x)|2, are plottedin Fig. WHY ARE complex numbers needed in quantum mechanics ?A. Due to the uncertainty principleThis argument is found in R. Shankar s textbook1for the introductory level and goes asfollows. Consider 1 particle in 1 dimension and assume de Broglie s matter waves relation( =2 ~p), which is previously justified by empirical evidence from the double-slit experimentwith electrons.

7 Since there appears to be a wave associated with the electron, it is reason-able to assume that there is a wave function (x)describingit(ignoretimedependency).Po stulate that the absolute square of this function (| (x)|2)givestheprobabilitydensity, , the likelihood to find the particle between the positionsxandx+ last piece needed for the argument is Heisenberg s uncertainty principle. It is pre-sented qualitatively in Shankar s textbook by thought experiments like the gamma-ray mi-croscope. One way to state the principle is to say that it is impossible to prepare a particlein a state in which its momentum and position (along one axis) are exactly known. In otherwords, the product of the uncertainties xand phas a lower bound and the principle canbe roughly expressed mathematically by x p> ~.Now, suppose the electron is in a state of definite momentum. How would the wavefunction for this state look like?

8 According to de Broglie s relation, there is a wavelengthassociated with it, thus, it seems plausible to assume a classical/real oscillating wave of theform p(x)=Acos2 x =Acospx~,(1)where p(x)denotesastateofdefinitemomentum. p(x), as well as| p(x)|2, are plottedin Fig. 1..By looking at the graph of| p(x)|2(probability distribution) one realizes that it is incontradictionwith the uncertainty principle. If you know the momentum of the electronexactly, you should have no information about its position. However, the graph of| p(x)|2is implying that there are regions where it is more likely to find the electron than others. Ifthe uncertainty principle were to be respected, Shankar argues, the probability distributionshould be flat , meaning that it is equally likely to find it anywhere. Thus, one cannot3II. WHY ARE complex numbers needed in quantum mechanics ?A. Due to the uncertainty principleThis argument is found in R.

9 Shankar s textbook1for the introductory level and goes asfollows. Consider 1 particle in 1 dimension and assume de Broglie s matter waves relation( =2 ~p), which is previously justified by empirical evidence from the double-slit experimentwith electrons. Since there appears to be a wave associated with the electron, it is reason-able to assume that there is a wave function (x)describingit(ignoretimedependency).Po stulate that the absolute square of this function (| (x)|2)givestheprobabilitydensity, , the likelihood to find the particle between the positionsxandx+ last piece needed for the argument is Heisenberg s uncertainty principle. It is pre-sented qualitatively in Shankar s textbook by thought experiments like the gamma-ray mi-croscope. One way to state the principle is to say that it is impossible to prepare a particlein a state in which its momentum and position (along one axis) are exactly known.

10 In otherwords, the product of the uncertainties xand phas a lower bound and the principle canbe roughly expressed mathematically by x p> ~.Now, suppose the electron is in a state of definite momentum. How would the wavefunction for this state look like? According to de Broglie s relation, there is a wavelengthassociated with it, thus, it seems plausible to assume a classical/real oscillating wave of theform p(x)=Acos2 x =Acospx~,(1)where p(x)denotesastateofdefinitemomentum. p(x), as well as| p(x)|2, are plottedin Fig. 1..By looking at the graph of| p(x)|2(probability distribution) one realizes that it is incontradictionwith the uncertainty principle. If you know the momentum of the electronexactly, you should have no information about its position. However, the graph of| p(x)|2is implying that there are regions where it is more likely to find the electron than others. Ifthe uncertainty principle were to be respected, Shankar argues, the probability distributionshould be flat , meaning that it is equally likely to find it anywhere.