Introduction To Quantum Mechanics

Introduction to quantum mechanics

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THE SCHRODINGER EQUATION

​deal of quantum mechanics already, whether you realize it or not. The outline of this chapter is as follows. In Section 10.1 we give a brief history of the development of quantum mechanics. In Section 10.2 we write down, after some motivation, the Schrodinger wave equation, both the time-dependent and time-independent forms. In Section 10.3 we discuss a number of examples. The most important thing to take away from this section is that all of the examples we discuss have exact analogies in the string/spring systems earlier in the book. So we technically won’t have to solve anything new here. All the work has been done before. The only thing new that we’ll have to do is interpret the old results. In Section 10.4 we discuss the uncertainty principle. As in Section 10.3, we’ll find that we already did the necessary work earlier in the book. The uncertainty principle turns out to be a direct consequence of a result from Fourier analysis. But the interpretation of this result as an uncertainty principle has profound implications in quantum mechanics.

​A BRIEF HISTORY

​1913 (Bohr): Niels Bohr stated that electrons in atoms have wavelike properties. This correctly explained a few things about hydrogen, in particular the quantized energy levels that were known. 1924 (de Broglie): Louis de Broglie proposed that all particles are associated with waves, where the frequency and wavenumber of the wave are given by the same relations we found above for photons, namely E = ¯hω and p = ¯hk. The larger E and p are, the larger ω and k are. Even for small E and p that are typical of a photon, ω and k are very large because ¯h is so small. So any everyday-sized particle with large (in comparison) energy and momentum values will have extremely large ω and k values. This (among other reasons) makes it virtually impossible to observe the wave nature of macroscopic amounts of matter. This proposal (that E = ¯hω and p = ¯hk also hold for massive particles) was a big step, because many things that are true for photons are not true for massive (and nonrelativistic) particles. For example, E = pc (and hence ω = ck) holds only for massless particles (we’ll see below how ω and k are related for massive particles). But the proposal was a reasonable one to try. And it turned out to be correct, in view of the fact that the resulting predictions agree with experiments. The fact that any particle has a wave associated with it leads to the so-called waveparticle duality. Are things particles, or waves, or both? Well, it depends what you’re doing with them. Sometimes things behave like waves, sometimes they behave like particles. A vaguely true statement is that things behave like waves until a measurement takes place, at which point they behave like particles. However, approximately one million things are left unaddressed in that sentence. The wave-particle duality is one of the things that few people, if any, understand about quantum mechanics. 1925 (Heisenberg): Werner Heisenberg formulated a version of quantum mechanics that made use of matrix mechanics. We won’t deal with this matrix formulation (it’s rather difficult), but instead with the following wave formulation due to Schrodinger (this is a waves book, after all). 1926 (Schrodinger): Erwin Schrodinger formulated a version of quantum mechanics that was based on waves. He wrote down a wave equation (the so-called Schrodinger equation) that governs how the waves evolve in space and time. We’ll deal with this equation in depth below. Even though the equation is correct, the correct interpretation of what the wave actually meant was still missing. Initially Schrodinger thought (incorrectly) that the wave represented the charge density. 1926 (Born): Max Born correctly interpreted Schrodinger’s wave as a probability amplitude. By “amplitude” we mean that the wave must be squared to obtain the desired probability. More precisely, since the wave (as we’ll see) is in general complex, we need to square its absolute value. This yields the probability of finding a particle at a given location (assuming that the wave is written as a function of x). This probability isn’t a consequence of ignorance, as is the case with virtually every other example of probability you’re familiar with. For example, in a coin toss, if you know everything about the initial motion of the coin (velocity, angular velocity), along with all external influences (air currents, nature of the floor it lands on, etc.), then you can predict which side will land facing up. Quantum mechanical probabilities aren’t like this. They aren’t a consequence of missing information. The probabilities are truly random, and there is no further information (so-called “hidden variables”) that will make things unrandom. The topic of hidden variables includes various theorems (such as Bell’s theorem) and experimental results that you will learn about in a quantum mechanics course.

1926 (Dirac): Paul Dirac showed that Heisenberg’s and Schrodinger’s versions of quantum mechanics were equivalent, in that they could both be derived from a more general version of quantum mechanics. 10.2 The Schrodinger equation In this section we’ll give a “derivation” of the Schrodinger equation. Our starting point will be the classical nonrelativistic expression for the energy of a particle, which is the sum of the kinetic and potential energies. We’ll assume as usual that the potential is a function of only x. We have E = K + V = 1 2 mv2 + V (x) = p 2 2m + V (x). (3) We’ll now invoke de Broglie’s claim that all particles can be represented as waves with frequency ω and wavenumber k, and that E = ¯hω and p = ¯hk. This turns the expression for the energy into ¯hω = ¯h 2 k 2 2m + V (x). (4) A wave with frequency ω and wavenumber k can be written as usual as ψ(x, t) = Aei(kx−ωt) (the convention is to put a minus sign in front of the ωt). In 3-D we would have ψ(r, t) = Aei(k·r−ωt) , but let’s just deal with 1-D. We now note that ∂ψ ∂t = −iωψ =⇒ ωψ = i ∂ψ ∂t , and ∂ 2ψ ∂x2 = −k 2ψ =⇒ k 2ψ = − ∂ 2ψ ∂x2 . (5) If we multiply the energy equation in Eq. (4) by ψ, and then plug in these relations, we obtain ¯h(ωψ) = ¯h 2 2m (k 2ψ) + V (x)ψ =⇒ i¯h ∂ψ ∂t = −¯h 2 2m · ∂ 2ψ ∂x2 + V ψ (6) This is the time-dependent Schrodinger equation. If we put the x and t arguments back in, the equation takes the form, i¯h ∂ψ(x, t) ∂t = −¯h 2 2m · ∂ 2ψ(x, t) ∂x2 + V (x)ψ(x, t). (7) In 3-D, the x dependence turns into dependence on all three coordinates (x, y, z), and the ∂ 2ψ/∂x2 term becomes ∇2ψ (the sum of the second derivatives). Remember that Born’s (correct) interpretation of ψ(x) is that |ψ(x)| 2 gives the probability of finding the particle at position x. Having successfully produced the time-dependent Schrodinger equation, we should ask: Did the above reasoning actually prove that the Schrodinger equation is valid? No, it didn’t, for three reasons. 1. The reasoning is based on de Broglie’s assumption that there is a wave associated with every particle, and also on the assumption that the ω and k of the wave are related to E and p via Planck’s constant in Eqs. (1) and (2). We had to accept these assumptions on faith. 2. Said in a different way, it is impossible to actually prove anything in physics. All we can do is make an educated guess at a theory, and then do experiments to try to show​

​INTRODUCTION TO QUANTUM MECHANICS

THE SCHRODINGER EQUATION

​THE SCHRODINGER EQUATION

In this section we’ll give a “derivation” of the Schrodinger equation. Our starting point will be the classical nonrelativistic expression for the energy of a particle, which is the sum of the kinetic and potential energies. We’ll assume as usual that the potential is a function of only x. We have E = K + V = 1 2 mv2 + V (x) = p 2 2m + V (x). (3) We’ll now invoke de Broglie’s claim that all particles can be represented as waves with frequency ω and wavenumber k, and that E = ¯hω and p = ¯hk. This turns the expression for the energy into ¯hω = ¯h 2 k 2 2m + V (x). (4) A wave with frequency ω and wavenumber k can be written as usual as ψ(x, t) = Aei(kx−ωt) (the convention is to put a minus sign in front of the ωt). In 3-D we would have ψ(r, t) = Aei(k·r−ωt) , but let’s just deal with 1-D. We now note that ∂ψ ∂t = −iωψ =⇒ ωψ = i ∂ψ ∂t , and ∂ 2ψ ∂x2 = −k 2ψ =⇒ k 2ψ = − ∂ 2ψ ∂x2 . (5) If we multiply the energy equation in Eq. (4) by ψ, and then plug in these relations, we obtain ¯h(ωψ) = ¯h 2 2m (k 2ψ) + V (x)ψ =⇒ i¯h ∂ψ ∂t = −¯h 2 2m · ∂ 2ψ ∂x2 + V ψ (6) This is the time-dependent Schrodinger equation. If we put the x and t arguments back in, the equation takes the form, i¯h ∂ψ(x, t) ∂t = −¯h 2 2m · ∂ 2ψ(x, t) ∂x2 + V (x)ψ(x, t). (7) In 3-D, the x dependence turns into dependence on all three coordinates (x, y, z), and the ∂ 2ψ/∂x2 term becomes ∇2ψ (the sum of the second derivatives). Remember that Born’s (correct) interpretation of ψ(x) is that |ψ(x)| 2 gives the probability of finding the particle at position x. Having successfully produced the time-dependent Schrodinger equation, we should ask: Did the above reasoning actually prove that the Schrodinger equation is valid? No, it didn’t, for three reasons. 1. The reasoning is based on de Broglie’s assumption that there is a wave associated with every particle, and also on the assumption that the ω and k of the wave are related to E and p via Planck’s constant in Eqs. (1) and (2). We had to accept these assumptions on faith. 2. Said in a different way, it is impossible to actually prove anything in physics. All we can do is make an educated guess at a theory, and then do experiments to try to show​

INTRODUCTION TO QUANTUM MECHANICS

​string can be the superposition of various normal modes with definite ω’s. The same reasoning applies here as with all the other waves we’ve discussed: From Fourier analysis and from the linearity of the Schrodinger equation, we can build up any general wavefunction from ones with specific energies. Because of this, it suffices to consider the time-independent Schrodinger equation. The solutions for that equation form a basis for all possible solutions.1 Continuing with our standard strategy of guessing exponentials, we’ll let ψ(x) = Aeikx . Plugging this into Eq. (9) and canceling the e ikx gives (going back to the ¯hω instead of E) ¯hω = − ¯h 2 2m (−k 2 ) + V (x) =⇒ ¯hω = ¯h 2 k 2 2m + V (x). (10) This is simply Eq. (4), so we’ve ended up back where we started, as expected. However, our goal here was to show how the Schrodinger equation can be solved from scratch, without knowing where it came from. Eq. (10) is (sort of) a dispersion relation. If V (x) is a constant C in a given region, then the relation between ω and k (namely ω = ¯hk2/2m + C) is independent of x, so we have a nice sinusoidal wavefunction (or exponential, if k is imaginary). However, if V (x) isn’t constant, then the wavefunction isn’t characterized by a unique wavenumber. So a function of the form e ikx doesn’t work as a solution for ψ(x). (A Fourier superposition can certainly work, since any function can be expressed that way, but a single e ikx by itself doesn’t work.) This is similar to the case where the density of a string isn’t constant. We don’t obtain sinusoidal waves there either. 10.3 Examples In order to solve for the wavefunction ψ(x) in the time-independent Schrodinger equation in Eq. (9), we need to be given the potential energy V (x). So let’s now do some examples with particular functions V (x). 10.3.1 Constant potential The simplest example is where we have a constant potential, V (x) = V0 in a given region. Plugging ψ(x) = Aeikx into Eq. (9) then gives E = ¯h 2 k 2 2m + V0 =⇒ k = s 2m(E − V0) ¯h 2 . (11) (We’ve taken the positive square root here. We’ll throw in the minus sign by hand to obtain the other solution, in the discussion below.) k is a constant, and its real/imaginary nature depends on the relation between E and V0. If E > V0, then k is real, so we have oscillatory solutions, ψ(x) = Aeikx + Be−ikx . (12) But if E < V0, then k is imaginary, so we have exponentially growing or decaying solutions. If we let κ ≡ |k| = p 2m(V0 − E)/¯h, then ψ(x) takes the form, ψ(x) = Aeκx + Ba−κx . (13) We see that it is possible for ψ(x) to be nonzero in a region where E < V0. Since ψ(x) is the probability amplitude, this implies that it is possible to have a particle with E < V0. 1The “time-dependent” and “time-independent” qualifiers are a bit of a pain to keep saying, so we usually just say “the Schrodinger equation,” and it’s generally clear from the context which one we mean.