How do I introduce Schrödinger equation during the quantum mechanics course?
aa r X i v : . [ phy s i c s . e d - ph ] O c t How do I introduce Schr¨odinger equation duringthe quantum mechanics course?
T. Mart
Departemen Fisika, FMIPA, Universitas Indonesia, Depok 16424, IndonesiaE-mail: [email protected]
Abstract.
In this paper I explain how I usually introduce the Schr¨odinger equationduring the quantum mechanics course. My preferred method is the chronological one.Since the Schr¨odinger equation belongs to a special case of wave equations I start thecourse with introducing the wave equation. The Schr¨odinger equation is derived withthe help of the two quantum concepts introduced by Max Planck, Einstein, and deBroglie, i.e., the energy of a photon E = ~ ω and the wavelength of the de Broglie wave λ = h/p . Finally, the difference between the classical wave equation and the quantumSchr¨odinger one is explained in order to help the students to grasp the meaning ofquantum wavefunction Ψ( r , t ). A comparison of the present method to the approachesgiven by the authors of quantum mechanics textbooks as well as that of the originalNuffield A level is presented. It is found that the present approach is different fromthose given by these authors, except by Weinberg or Dicke and Wittke. However, theapproach is in line with the original Nuffield A level one. Keywords : quantum mechanics, wave equation, Schr¨odinger equation, de Broglie wave,neutron diffraction30 October 2020
1. Introduction
Quantum mechanics is a notoriously difficult and abstract topic. I remember that whenI started my first year of my undergraduate study more than 30 years ago, my friend toldme that in the following years we were going to hear about Schr¨odinger equation whichis elegant but complicated and difficult. I was very excited at that time and felt thatI could not wait any longer to attend the quantum mechanics course. However, beforeI could acquaint this Schr¨odinger masterpiece, I accidentally entered the Introductionto Solid State Physics course, where I had to calculate the probability of an electrontransition represented by the Dirac bracket that sandwiches a potential operator betweentwo wavefunctions. Immediately, I got the opinion that quantum mechanics was verydifficult, not interesting, and I had even no idea why should people write such a bracket.It is not an exaggeration if Feynman once said in his well known quote: ”I thinkI can safely say that nobody understands quantum mechanics [1].” Of course what ow do I introduce Schr¨odinger equation
2. Introducing the classical wave equation A ( r ) r (a) (b) Figure 1. (a) Example of a very intuitive wave phenomenon, but mathematically aslightly complicated one. (b) The one-dimensional cut of this wave shows that theoscillation of this wave is damped.
Since the Schr¨odinger equation is a differential wave equation, it is essential toremind the students about the wave equation in the classical mechanics. However, itshould be remembered that during that time not all students can quickly understandthe wave equation in terms of sinusoidal function or differential equation. Therefore,I usually start with the very intuitive wave phenomenon depicted in Fig. 1(a). Thisphenomenon can be observed if we, e.g., drop a stone in the water. Since the propagationof the wave is radial and the amplitude decreases as the radius increases, the simpleststationary wavefunction reads A ( r ) = A sin rr , (1) ow do I introduce Schr¨odinger equation − A A π π π π π π A ( x , t ) kx or −ω t Figure 2.
A mathematically simple wave phenomenon, but less intuitive one, i.e., A ( x, t ) = A sin( kx − ωt ). where A ( r ) is the displacement of the water at the position r and A indicates theamplitude at r = 0. Note that the wavefunction of the water ripples is actually notstationary, it is a function of time and although we only consider the stationary casegiven by Eq. (1) it is still not simple. It has a dependence on r in the denominatorwhich is indeed important to suppress the amplitude in the location far from the wavecenter, as clearly shown by Fig. 1(b).A mathematically much simpler wave equation is the sine wave displayed in Fig. 2.Example of the sine wave is the standing wave observed in the string of a guitar.However, since everybody can easily drop a stone in the water, whereas not everybodyhas an access to a guitar, the second example is less intuitive. Nevertheless, all studentshave learned standing wave in high school and since the wave is very simple, we willstart our calculation with the sine wave.The wavefunction displayed in Fig. 2 can be written as A ( x, t ) = A sin( kx − ωt ) , (2)where k is the wave number and ω is the angular frequency. A computer software like Mathematica or Matlab can help the lecturer to simulate this wave during the course.Note that the two parameters define the phase velocity of the wave, i.e., v = ω/k . (3)By taking the second derivatives of Eq. (2) with respect to x and t and using Eq. (3) itis easy to show that ∂ A ( x, t ) ∂t = v ∂ A ( x, t ) ∂x . (4)Equation (4) is the wave differential equation in classical mechanics. All parameters andvariables given in Eq. (4) are real and physical. For instance, the displacement A ( x, t )is real and can be observed. For the three dimensional problem this equation is givenin the form of the Laplacian operator ∇ , i.e., ∂ A ( r , t ) ∂t = v ∇ A ( r , t ) . (5) ow do I introduce Schr¨odinger equation i = √− A ( x, t ) = A e i ( kx − ωt ) . (6)The displacement A ( x, t ) is still observable and real by keeping in mind that the observedquantities are represented by the real and imaginary parts of A ( x, t ). Here, the imaginarynumber i works as a unit vector pointing to completely different direction from the realone. Thus, the i number completely separates the sine and cosine waves.
3. Introducing quantum concepts to obtain the Schr¨odinger equation
Before introducing the Schr¨odinger equation it is important to explain to students whySchr¨odinger needed this equation at that time. Therefore, a brief chronological historyof this equation is strongly advocated.We begin with the finding of Max Planck in 1900 which was then proven by Einsteinin 1905 that the energy of electromagnetic wave is carried out by the quanta (packets)called photons. Each quantum carries energy of E = h ν = ~ ω , (7)where h and ν are the Planck constant and the frequency of the electromagnetic wave,respectively, whereas ~ = h/ π . Equation (7) indicates the matter property of awave, since by using the Einstein energy momentum relation E = p c + m c andremembering that the mass of rest photon is zero, we obtain from Eq. (7) that pc = hc/λ or the (matter) momentum of the photon reads p = hλ , (8)where we have used the fact that the speed of electromagnetic wave c = λν .In 1924 Louis de Broglie had an idea that in addition to the photon theory of MaxPlanck and Einstein all particles with momentum p , including the massive electrons,have also a wave property with the wavelength of λ = hp . (9)At first glance Eq. (9) is nothing but Eq. (8). Nevertheless, de Broglie wrote a 73 pagesthesis to defense this idea [4]. Even the thesis committee did not know what to towith the thesis, so they sent it to Einstein. Fortunately Einstein saw the point andrecommended the approval. Einstein sent the thesis to his colleagues he knew would beinterested in the de Broglie’s idea [5].On November 1925 Erwin Schr¨odinger attended a colloquium held by his colleaguePeter Debye at the E. T. H. Zurich. As noticed by Felix Bloch in his talk given atthe 1976 American Physical Society meeting [6] he heard that during the colloquium ow do I introduce Schr¨odinger equation k = 2 π/λ , we obtain k = 2 π ph = p ~ . (10)Thus, with the help of Eq. (8) we can recast Eq. (6) into the formΨ( x, t ) = Ψ e i ( px − Et ) / ~ , (11)where we have replaced the displacement A ( x, t ) with Ψ( x, t ), along with the amplitude,to indicate that we have included the quantum concept introduced by Max Planck,Einstein, and de Broglie, given by Eqs. (7) and (9), in the wavefunction. In fact,Schr¨odinger introduced the Ψ function in his equation and called it “a new unknownfunction” in his 1926 paper [7].Now, since we are talking about a non-relativistic free particle described by a planewave, the total energy of the particle is only kinetic energy, i.e., E = p / m . By usingthis fact and taking the first derivative of the wavefunction given in Eq. (11) with respectto t , as well as the second derivative with respect to x , we obtain i ~ ∂ Ψ( x, t ) ∂t = − ~ m ∂ Ψ( x, t ) ∂x , (12)which is the wave equation for a free particle with mass m we are looking for. Since theright hand side of Eq. (12) corresponds to the kinetic energy T , we may generalize thisequation to describe a particle moving under the influence of a potential energy V andin three dimensional coordinate system as i ~ ∂ Ψ( r , t ) ∂t = − ~ m ∇ Ψ( r , t ) + V ( r ) Ψ( r , t ) , (13)where we have used the fact that the total energy E = T + V . Equation (13) is theSchr¨odinger equation in general form. We note that similar derivations can be alsofound in the literature, e.g., Refs. [8, 9, 10, 11].
4. Comparing classical mechanics to quantum mechanics
By introducing the Schr¨odinger equation in Section 3 we have jumped to the quantumworld which is very different from the classical one. At this stage it is important toexplain to the students the difference between the classical wave equation given byEq. (5) and its quantum counterpart given by Eq. (13). ow do I introduce Schr¨odinger equation A ( r , t ) is physical and observable. It is easy toimagine that the water ripples shown in Fig. 1, or the string oscillation displayed inFig. 2, can be represented by the variation of A ( r , t ) due to the variations of r and t .On the contrary, Ψ( r , t ) given in Eq. (13) is not a displacement. It is just a wavefunctiondescribing the de Broglie wave given by Eq. (9).Is the de Broglie wave physical? Obviously, the answer is no. The massive particlesexhibit the wave phenomenon, but the wave itself is not physical. To comprehend thiswe have to go back to the experimental verification of Eq. (9). The first experimentwas performed by Davisson and Germer in 1927 by using electron scattering on a singlenickel crystal [12]. A more modern experiment using cold neutrons diffraction on singleand double slits was performed by Zeilinger and his collaborators in Grenoble [13]. Inprinciple, it is similar to the conventional diffraction experiment. However, instead ofusing real photons (visible light) here one uses neutron beams and the general diffractionpattern can be reproduced. Therefore, we may conclude that after passing the slits theneutrons exhibit one of the wave phenomena, i.e., diffraction. This clarifies that we donot detect the de Broglie wave, but we merely observe its phenomenon.As a consequence, the displacement A ( r , t ) is real, whereas its counterpart Ψ( r , t )is allowed to be complex function. The latter was actually problematic, even forSchr¨odinger himself, because he did not expect it. In 1926 Max Born came up with theidea that Ψ( r , t ) is a probability amplitude and its modulus squared is the probabilitydensity [15], which was considered by Schr¨odinger with great doubt [6].The second difference is, unlike the Schr¨odinger equation, the wave equation issymmetric in the order of derivatives. This explains why the Schr¨odinger equation isnon-relativistic, since it cannot be formulated in a covariant form as in the case of thewave equation for the photons. We know that the covariant formulation requires thatboth space and time terms should be in the same order.
5. Comparison with other approaches
To the best of my knowledge, there has been no explicit discussion in the literature onthe most effective method to introduce the Schr¨odinger equation during the courseof quantum mechanics. However, since most of quantum mechanics textbooks arewritten based on the teaching experience of the authors, we can compare my approachexplained here with the steps used by these authors before they write the Schr¨odingerequation. Furthermore, the Schr¨odinger equation is usually introduced in the first orsecond chapter of quantum mechanics textbooks. Thus, their approaches should beeasily identified.Let us start with the book of Gasiorowicz [14] which has been considered as oneof the standard quantum mechanics textbooks used in most universities for relativelylong time, i.e., since its first edition in 1974 up to now. In the Sakurai’s book ofModern Quantum Mechanics the editor San Fu Tuan wrote that Gasiorowicz is well ow do I introduce Schr¨odinger equation H Ψ = E Ψ, with H the Hamiltonian of the system. Althoughmost of the books start with historical background of quantum mechanics, it is given ina separate chapter before the one where the Schr¨odinger equation is introduced. Thereseems to be an unwritten consensus among these authors that students should havealready known the Schr¨odinger equation as in the case of classical mechanics, in whichstudents have been already very familiar with the Newton laws.Among the recent quantum mechanics books, only the one written by StevenWeinberg is different [11]. Weinberg opens the book with the first chapter discussingthe historical introduction, which is constructed in a chronological order. Indeed, in thebeginning of the first chapter Weinberg wrote: ”The principles of quantum mechanicsare so contrary to ordinary intuition that they can best be motivated by taking a lookat their prehistory.” Thus, in the Weinberg’s book the Schr¨odinger equation is deriveddirectly in Chapter 1, in a similar manner as the approach given in the present paper. ow do I introduce Schr¨odinger equation • photons, wave-particle duality, • electrons, electrons as a wave, • waves in boxes, Sch¨odinger’s equation, • the scope of wave mechanics.From the ordering of the four items above we may conclude that, in principle, thediscussion of Sch¨odinger equation in the original Nuffield A level is in line with myapproach explained in this paper.
6. Further consideration
I agree with Steven Weinberg that the best way to introduce quantum mechanics is byconsidering its chronological history [11]. This include the Schr¨odinger equation, whichis the main equation in non-relativistic quantum mechanics. Nevertheless, to obtain amore objective result a more quantitative investigation should be performed in the class.This can be carried out in two parallel student groups, where two different approachescan be applied. Such an investigation is possible in my physics department, since eachsemester it runs two or three parallel classes for the Introductory Quantum Mechanicscourse. Therefore, it is my plan to study the effectiveness of the present approach inthe near future.Having introduced the Schr¨odinger equation, the next step is presenting theapplications of this equation, i.e., variations of the potential V in Eq. (13). Of course, thebest example is the application for the Hydrogen atom as described in the Schr¨odingerpaper [7]. However, since simpler is better, I like to start with the traditional approach,i.e., one-dimensional potential. My favorite case here is the simple potential barrier andpotential well, which have a wide range of applications, i.e., from scanning tunnelingmicroscope to the nuclear reactions in stars [14]. Such applications have naturally strongimpact on the student motivation to attend the course.
7. Summary and Conclusion
I have discussed that introducing the Schr¨odinger equation during the quantummechanics course is a critical step, since it constitutes a transition from classical ow do I introduce Schr¨odinger equation
Acknowledgment
The author was partly supported by the 2020 PUTI Q2 Research Grant of UniversitasIndonesia, under contract No. NKB-1652/UN2.RST/HKP.05.00/2020.
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