PY 351 Modern Physics - Lecture notes, 1

PY 351 Modern Physics - Lecture notes, 1 Copyright by Claudio Rebbi, Boston University, October 2016, October 2017. These notes cannot be duplicated and distributed without explicit permission of the author. 1. History Important note about sources: historical references have been drawn from Wikipedia: (https://www.wikipedia.org/) and for the equation of Hamilton-Jacobi also from Scholarpedia: (http://www.scholarpedia.org/article/main Page). The passages taken from these sources appear in italics, although the text has generally been edited to adapt it to the context of the lecture. Hans Geiger and Ernest Marsden performed experiments in 1908, 1909, 1910, 1913 to measure the scattering of alpha particles (discovered earlier by Rutherford) to measure the deflection caused by passage through matter (typically metal foils.) In 1911 Ernest Rutherford published a paper with the interpretation of the scattering as due to a positively charged nucleus inside the atoms. This gave origin to the understanding of atoms as made of a very small (relative to the atom s size) positively charged nucleus surrounded by orbiting electrons. Major problem: according to the classical equations of mechanics and electromagnetism the electrons would radiate electromagnetic waves, lose energy and fall onto the nucleus, and the atom would collapse. In 1913 Niels Bohr proposed a model which avoided the collapse of the atom and also produced good agreement with the energy levels of the hydrogen atom determined from spectroscopic data. 2. Bohr s model We first solve the classical equation of motion for a circular orbit of radius r. The potential energy of the electron is e2 V 4πɛ 0 r where e is the electron s charge. Correspondingly the attractive force is (in magnitude) F (1) e2 4πɛ 0 r 2 (2) which must match the magnitude of the electron s masse m times the centripetal acceleration mv 2 r F e2 4πɛ 0 r 2 (3) 1

From this equation we could derive the value of the velocity v along the orbit as function of r. But Bohr s model is based on a hypothesis about the electron s angular momentum l mrv. Consequently it is more convenient to multiply Eq. 3 by mr 3 to get m 2 r 2 v 2 l 2 me2 r 4πɛ 0 (4) Bohr postulates that the angular momentum of the electron can only take values equal to an integer multiple of h/2π, a constant which is denoted by h, i.e. l n h nh 2π We thus get me 2 r n 2 h 2 (6) 4πɛ 0 where the integer n will eventually referred to as the principal quantum number. This restricts the possible radii of the orbits to the values The quantity (5) r n 4πɛ 0 h 2 me 2 n 2 (7) a 0 4πɛ 0 h 2 0.0529nm (8) me 2 is called the Bohr radius. So in Bohr s model the possible values radius of the electron s orbit are given by n 2 times a 0. About the corresponding energy levels, if we multiply Eq. 3 by r/2 we find mv 2 e2 (9) 2 8πɛ 0 r i.e. the kinetic energy K on the electron in the orbit is one-half of the magnitude of the potential energy. This gives for the energy levels E n V n + K n e2 + e2 e2 me4 1 4πɛ 0 r n 8πɛ 0 r n 8πɛ 0 r n 8πɛ 0 h 2 n 13.6eV 1 (10) 2 n 2 The constant is called the Rydberg energy. 3. Wave-like nature of matter Ry me4 2 13.6eV (11) 8πɛ 0 h In 1924 Louis de Broglie postulated that a massive particle with momentum p should have a corresponding matter wave. The wavelength would be related to the particle s momentum by λ h p (12) 2

or, equivalently, the particle would be associated with a wave of the form ψ(x) Ae ıpx/ h (13) (x λ gives exp(ıpλ/ h) exp(ıph/p h) exp(2πı) 1.) Note the analogy with the photon s wavelength λ c f hc hf hc E h (14) p since for a photon E pc. Note that in classical mechanics the angular momentum l is related to rotation angle φ in the same way as linear momentum p is related to the x-coordinate. This leads us to associate to l a wave (around the rotation circle) of angular-length φ h l (15) (Remember that h and l have the same dimensions, so h/l is dimensionless, like an angle.) Bohr s quantization condition l nh/2π (see Eq. 5) gives us then φ h l h nh/(2π) 2π n (16) which states that the angular-length of the wave is an integer fraction of 2π, exactly as needed to fit an integer number of wave-lengths into the circle. Thus Bohr s condition may be seen as the requirement that the matter-wave associated with the electron s motion forms a standing wave over the orbit. 4. More history During the years 1923-1927 Clinton Davisson and Lester Germer performed experiments by scattering electrons off crystals which validated the hypothesis that electrons are associates with some form of wave. In 1925 Erwin Schrödinger postulated that matter waves obey a linear differential equation, which he published in 1926. In the period 1830-1832 Sir William Rowan Hamilton developed an equation for the propagation of optical rays in media with varying refraction index. Later, in 1842-1843 Carl Gustav Jacob Jacobi perfected the equation and showed that it can be used as an alternative to the evolution equations of classical mechanics. The Hamilton-Jacobi equation of classical mechanics, well known to physicist struggling to develop quantum mechanics, provided a background for the development of the Schrödinger equation. 3

5. The optical ray limit of the wave-equation. Consider the wave equation 2 φ(x, t) v 2 2 φ(x, t) (17) 2 2 With constant v the equation admits (among others) the solution φ(x, t) Ae ıω(x/v t) (18) which describes a wave of constant amplitude A, frequency ν ω/(2π), and wave-length λ v/ν, propagating in the forward x direction with velocity v. Equation 17 becomes much more difficult to solve if v, instead of being a constant, is allowed to vary v v(x) (19) i.e. in presence of an index of refraction which varies with x. In this case, for general v(x) the equation can only be solved numerically. However it is amenable to an analytic solution in the limit where the local wave-length, which we define as λ(x) v(x) ν 2πv(x) ω is much smaller than the length over which v(x) exhibits an appreciable variation. From Eq. 20 we see that this limit can also be seen as the limit where ω. In the case of electromagnetic waves, the limit is called the optical ray limit. In order to find the optical ray solution to the wave equation, we write φ(x, t) in the form (20) φ(x, t) e ıωs(x,t) (21) Where S(x, t) is a function, to be determined, which we will call the light-front function (that is my pedagogical term, it is actually called the action.) Substituting the expression for φ given by Eq. 21 into Eq. 17, with simple algebra and neglecting terms of order less than ω 2, one finds the following equation for S(x, t) ( ) 2 ( v(x) This equation may be solved by separation of variables: we set Substituting into Eq. 22 we find ) 2 (22) S(x, t) S x (x) S t (t) (23) ( dst (t) ) 2 ( v(x) ds ) x(x) 2 (24) dt 4

But since the l.h.s. of this equation depends only on t while the r.h.s. depends only on x, both must be equal to a constant, which we can take without loss of generality to be equal to 1. This we get the two equations for S t (t) and S x (x) ( dst (t) ) 2 1 (25) dt and ( v(x) ds ) x(x) 2 1 (26) Or, taking the square roots with the positive sign (a negative sign in either of the two would produce a backward propagating wave) and ds t (t) dt 1 (27) v(x) ds x(x) 1 (28) Let us choose the origin of the t and x axes so that S t (0) S x (0) 0. Equation. 27 is then trivially solved by S t (t) t (29) Equation 28 requires a little more consideration. It can be expressed as ds x v(x) (30) Let us define the function of x τ(x) x 0 v(x ) Notice that τ(x) is the time it gets to a particle moving with velocity v(x) to get from x 0 to x. Then the solution of Eq. 28 is (31) S x (x) τ(x) (32) Putting everything together, we have found the optical ray solution φ(x, t) e ıωs(x,t) e ıω(sx(x) St(t)) e ıω(τ(x) t) (33) Notice that in the solution with constant v, φ(x, t) exp(ıω(x/v t), x/v also represents the time it would have taken to a particle moving with velocity v to get from x 0 to x. So, we may think of a light-front with φ(x, t) 1, for example the one which at t 0 is at x 0, as moving like a particle which will get at x in time t τ(x). However we should not think of these light-particles as photons. This is still classical physics: we are in the 5

years 1830-1832. We have simply found an alternative description of light rays with slowly varying refraction index where the waves behave as particles. 6. The equation of Hamilton-Jacobi and the Schrödinger equation. The equation of Hamilton-Jacobi is a non-linear wave equation. With our current understanding the equation of Hamilton-Jacobi can be seen as the optical limit (limit for small wave-lengths or, equivalently, h 0) of the Schrödinger equation (and we will later derive it this way), but Schrödinger went the opposite way and found a linear wave equation which would reduce to the Hamilton-Jacobi and thus describe classical motion in the limit h 0. In the Hamiltonian formulation of classical mechanics one expresses the energy E of a system in terms of the coordinates of the system s components and the corresponding momenta. The function which gives the energy in terms of coordinates and momenta is called the Hamiltonian and is generally denoted by H. With a single particle of mass m moving along the x-axis with momentum p and potential energy V (x) the Hamiltonian function is given by H(p, x) p2 + V (x) (34) 2m The equation of Hamilton-Jacobi describes the classical motion of the particle in a way similar to how one can describe the deflection of light-rays by a medium of variable index of refraction. The equation is obtained by replacing p in the Hamiltonian by / and by replacing E in the relation E H(p, x) (35) by /, which produces the non-linear pde (partial derivatives equation) ( H ), x 1 2m ( ) 2 + V (x) (36) (For the moment this must sound quite mysterious, but we will derive this equation from the Schrödinger equation by going to the classical limit, and it will be quite straightforward.) Schrödinger was looking for a linear wave equation which could replace Eq. 36 and still give origin to Eq. 36 in the classical limit. He was looking for a wave equation because the notion that at the quantum level particles exhibited some type of wave-like behavior was gradually getting hold, and the equation had to be a linear pde because linear pde obey a superposition principle (any linear combination of two solutions is also a solution), which is needed to generate interference patterns, and also because, in suitable conditions, they give origin to a discretized spectrum standing waves. Schrödinger s prescription was to start from the Hamiltonian function and to replace the classical momentum p with the differential operator ˆp (37) (We follow the convention of using a caret to differentiate operators from variables.) This 6

produces the differential operator Ĥ 1 ( ) ( 2m ) + V (x) h2 2m 2 + V (x) (38) 2 This differential operator would then act on a wave function ψ(x, t) and correspond to the operation performed in a measurement of energy. In this way the Hamiltonian operator Ĥ becomes the energy operator of the quantum mechanical system. The time evolution of the wave function is then obtained by replacing E in Eq. 35 with the differential operator /: E One thus obtains the Schrödinger wave equation ψ(x, t) (39) h2 2 ψ(x, t) Ĥψ(x, t) + V (x)ψ(x, t) (40) 2m 2 for the evolution of the Schrödinger wave function ψ(x, t) which describes the quantum system. 7. Properties and meaning of the wave function. The wave function should be continuous with continuous derivatives because the action of ˆp and Ĥ should be well defined. It should also be normalizable, i.e. the integral I(ψ) ψ(x, t) 2 ψ(x, t) ψ(x, t) (41) should be finite, because this is necessary for the probabilistic interpretation of ψ. possible then to normalize ψ, dividing it by I, in such a way that It is ψ(x, t) 2 1 (42) (One may encounter exceptions to the above rules, but they should be better considered as limit cases of situations where the rules are observed.) About the interpretation of ψ the most straightforward is that, if the wave-function is normalized to 1 (i.e. if Eq. 42 is satisfied), then ψ(x, t) 2 gives the probability density for finding the particle at a certain position x. More precisely, the probability that a measurement find the particle between x a and x b > a is given by p(a, b) b a ψ(x, t) 2 (43) Note that, unless otherwise specified, in what follows we will assume that the wave-function is normalized to 1. Incidentally, the normalization is preserved by the time evolution, as may be seen as follows: di(ψ) dt ( ψ(x, t) ψ(x, t) + ψ(x, t) 7 ψ(x, t) ) (44)

or, using Eq. 40 di(ψ) [( Ĥψ(x, t) ) ψ(x, t) + ψ(x, t) Ĥψ(x, t) ] dt [( 2 x ψ(x, t)/(2m) + V (x)ψ(x, t) ) ψ(x, t) ( +ψ(x, t) 2 x ψ(x, t)/(2m) + V (x)ψ(x, t) )] [( 2 x ψ(x, t) /(2m) V (x)ψ(x, t) ) ψ(x, t) ( +ψ(x, t) 2 x ψ(x, t)/(2m) + V (x)ψ(x, t) )] 1 [ 2 2m xψ(x, t) ψ(x, t) ψ(x, t) ψ(x, 2 t)] (45) which vanishes as can be seen by integrating by parts. But the interpretation of ψ(x, t) goes way beyond that. The wave function represents the quantum mechanical state of the system and can be used to evaluate the outcome of the measurement of any observable quantity. An observable quantity or simply observable is something that can be measured, like the position, momentum, or energy of the particle. Observables are represented by Hermitian linear operators which act on the wave function. The attribute Hermitian will be explained later. Let us denote one such observable and the corresponding operator by Ô. We do not need to distinguish between the observable and the operator. The outcome of the measurement of Ô is not certain. The measurement can produce different results (we could say because the measurement perturbs the system), but if we perform the same experiment many times and each time measure the same observable, then the average value or expectation value of the results will be given by O ψ(x) Ôψ(x) (46) (We left out the time dependence of ψ for simplicity, but as ψ evolves with time the expectation value of the measurement may also change.) In order to clarify these concepts let us consider the position x of the particle. x itself can be considered as a linear operator ˆx acting on ψ(x): ˆxψ(x) xψ(x) (47) Equation 47 defines a linear operation on ψ(x). The result of the action of ˆx on ψ(x) is the new function φ(x) xψ(x). The action is linear because if we consider a linear combination ψ(x) c 1 ψ 1 (x) + c 2 ψ 2 (x) (48) 8

by the very definition of Ô we will have ˆx ψ(x) x ψ(x) x(c 1 ψ 1 (x) + c 2 ψ(2))) c 1 xψ 1 (x) + c 2 xψ 2 (x) c 1ˆxψ 1 (x) + c 2ˆxψ 2 (x) (49) showing that the operation is linear. Following Eq. 46 the expectation value of ˆx, i.e. the average of many measurements of x is x ψ(x) xψ(x) x ψ(x, t) 2 (50) in agreement with the probabilistic interpretation of ψ(x, t) 2. Note, however, that ψ is a complex number, which we can write ψ(x, t) ψ(x, t) e ıφ(x,t) (51) and the angle φ, which we will call the phase of the wave function is as important as its magnitude ψ(x, t). Indeed, one could argue that the phase of ψ is more important than its magnitude. 8. The momentum operator. Another important operator is the momentum ˆp d The expectation value of the particle s momentum is given by p ψ(x) ( d ) ψ(x) ψ(x) dψ(x) (52) (53) and the phase of ψ enters in a crucial manner in this equation. Indeed, we cannot write the r.h.s. of Eq. 53 as ( d ) ψ(x) 2 (54) which would be an egregious error: among others, the expectation value of the momentum would turn out to be imaginary! And this brings us to the word Hermitian, which we used above without explanation. An operator Ô is called Hermitian if the expression ψ(x) Ôψ(x) (55) is real for any ψ(x), which is an obvious requirement if the expression is to represent the expectation value of the observable associated with Ô. Let us verify that ˆp is Hermitian. Consider the integral I ψ(x) dψ(x) (56) 9

and take its complex conjugate I ψ(x) dψ(x) (57) Integrating by parts and using the fact that ψ(x) vanishes at infinity we get I dψ(x) ψ(x) (58) But this is the original expression for I and so we conclude that I I and therefore that I is real. Note the relevance of the imaginary unit in the definition of ˆp. Without it the integral would be purely imaginary. An interesting question is what happens to the wave function after a measurement. To explore this point let us consider the linear operator ˆP (a, b), with a < b, which acts as follows: applied to a function ψ(x) it gives 0 for x < a or x > b and ψ(x) if a x b, i.e. ˆP (a, b)ψ(x) ξ(x) (59) with ξ(x) ψ(x) for a x b and 0 outside of this interval. The expectation value of the observable P (a, b) is P (a, b) ψ(x) ˆP (a, b)ψ(x) ψ(x) ξ(x) b a ψ(x) 2 (60) So expectation value of P (a, b) is the probability that the particle be found between a and b. However this is not the outcome of a measurement of P (a, b). The outcome of the measurement will be 1 or 0: 1 means that the particle can be localized between a and b, 0 otherwise. Of course, if we repeat the measurement many times and average the 1 s and 0 s we will find the expectation value given by Eq. 60, but this is not what we are now interested about. We consider just a single measurement, and this will give 1 or 0. So, suppose that we measure P (a, b) and get 1. What happens afterward? What happens is that ξ, normalized to 1, will become the new wave function of the particle. Namely, its wave function will become ψ(x) b a ψ(x) 2 for a x b, 0 otherwise (61) And we see here a major point of quantum mechanics: in general a measurement affects the state of the system, which may and often will emerge from the measurement with a different wave function, i.e. in a different state. 9. The equation of Hamilton-Jacobi We conclude this section by deriving the equation of Hamilton-Jacobi from the Schrödinger equation. The starting point is to express the complex wave function in terms of its magnitude A(x, t) and phase φ(x, t): ψ(x, t) A(x, t)e ıφ(x,t) (62) 10

In the classical limit h 0 the phase oscillates extremely rapidly, so we make this explicit by setting S(x, t) φ(x, t) (63) h The function S(x, t) is called the action and has dimensions of angular momentum. With this ψ takes the form ψ(x, t) A(x, t)e ıs(x,t)/ h (64) We substitute into the Schrödinger equation to obtain ψ(x, t) [A(x, t)eıs(x,t)/ h ] h2 2m Taking the derivatives this becomes [ [ h2 2 A(x, t) + 2 2m 2 +A(x, t) ( ı h h2 2 ψ(x, t) + V (x)ψ(x, t) (65) 2m 2 A(x, t) 2 [A(x, t)e ıs(x,t)/ h ] 2 + V (x)a(x, t)e ıs(x,t)/ h (66) + A(x, t) ī h A(x, t) ı h ) 2 ] e ıs(x,t) h ] e ıs(x,t) h + A(x, t) ī 2 S(x, t) h 2 + V (x)a(x, t) e ıs(x,t) h (67) We multiply by exp( ıs(x, t)/ h) and notice that, since both A(x, t) and S(x, t) are real functions, t he real and imaginary parts of the equation must match, we get two separate coupled, non-linear differential equations for A(x, t) and S(x, t): A(x, t) h2 2 A(x, t) + 1 A(x, t) 2m 2 2m ( ) 2 + V (x)a(x, t) (68) A(x, t) 1 A(x, t) 1 m 2m A(x, A(x, t) t) 2 (69) 2 We go to the classical limit by letting h 0 in these two equations. Equation 68 reduces then to an equation for S(x, t) only, namely ( ) 1 2 + V (x) (70) 2m while the other equation, which does not contain any explicit dependence on h, can be taken as an equation for A(x, t) once Eq. 36 has been solved for S. Equation 70, well known to physicists at the time of Schrödinger, is the equation of Hamilton-Jacobi which represents an alternative way to describe classical motion. We have seen it here emerge as the equation for the optical limit (limit of very short wave-length) of the Schrödinger equation. 11