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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£6.20
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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
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Quantum mechanics homework help 8
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Quantum Mechanics Homework Help 8
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 ...
Introduction to quantum
mechanics
David Morin, morin@physics.harvard.edu
This chapter gives a brief introduction to quantum mechanics. Quantum mechanics can be
thought of roughly as the study of physics on very small length scales, although there are
also certain macroscopic systems it directly applies to. The descriptor “quantum” arises
because in contrast with classical mechanics, certain quantities take on only discrete values.
However, some quantities still take on continuous values, as we’ll see.
In quantum mechanics, particles have wavelike properties, and a particular wave equa-
tion, the Schrodinger equation, governs how these waves behave. The Schrodinger equation
is different in a few ways from the other wave equations we’ve seen in this book. But these
differences won’t keep us from applying all of our usual strategies for solving a wave equation
and dealing with the resulting solutions.
In some respect, quantum mechanics is just another example of a system governed by a
wave equation. In fact, we will find below that some quantum mechanical systems have exact
analogies to systems we’ve already studied in this book. So the results can be carried over,
with no modifications whatsoever needed. However, although it is fairly straightforward
to deal with the actual waves, there are many things about quantum mechanics that are a
combination of subtle, perplexing, and bizarre. To name a few: the measurement problem,
hidden variables along with Bell’s theorem, and wave-particle duality. You’ll learn all about
these in an actual course on quantum mechanics.
Even though there are many things that are highly confusing about quantum mechanics,
the nice thing is that it’s relatively easy to apply quantum mechanics to a physical system
to figure out how it behaves. There is fortunately no need to understand all of the subtleties
about quantum mechanics in order to use it. Of course, in most cases this isn’t the best
strategy to take; it’s usually not a good idea to blindly forge ahead with something if you
don’t understand what you’re actually working with. But this lack of understanding can
be forgiven in the case of quantum mechanics, because no one really understands it. (Well,
maybe a couple people do, but they’re few and far between.) If the world waited to use
quantum mechanics until it understood it, then we’d be stuck back in the 1920’s. The
bottom line is that quantum mechanics can be used to make predictions that are consistent
with experiment. It hasn’t failed us yet. So it would be foolish not to use it.
The main purpose of this chapter is to demonstrate how similar certain results in quan-
tum mechanics are to earlier results we’ve derived in the book. You actually know a good
1
,2 CHAPTER 10. INTRODUCTION TO QUANTUM MECHANICS
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.
10.1 A brief history
Before discussing the Schrodinger wave equation, let’s take a brief (and by no means com-
prehensive) look at the historical timeline of how quantum mechanics came about. The
actual history is of course never as clean as an outline like this suggests, but we can at least
get a general idea of how things proceeded.
1900 (Planck): Max Planck proposed that light with frequency ν is emitted in quantized
lumps of energy that come in integral multiples of the quantity,
E = hν = h̄ω (1)
where h ≈ 6.63 · 10−34 J · s is Planck’s constant, and h̄ ≡ h/2π = 1.06 · 10−34 J · s.
The frequency ν of light is generally very large (on the order of 1015 s−1 for the visible
spectrum), but the smallness of h wins out, so the hν unit of energy is very small (at least on
an everyday energy scale). The energy is therefore essentially continuous for most purposes.
However, a puzzle in late 19th-century physics was the blackbody radiation problem. In a
nutshell, the issue was that the classical (continuous) theory of light predicted that certain
objects would radiate an infinite amount of energy, which of course can’t be correct. Planck’s
hypothesis of quantized radiation not only got rid of the problem of the infinity, but also
correctly predicted the shape of the power curve as a function of temperature.
The results that we derived for electromagnetic waves in Chapter 8 are still true. In
particular, the energy flux is given by the Poynting vector in Eq. 8.47. And E = pc for
a light. Planck’s hypothesis simply adds the information of how many lumps of energy a
wave contains. Although strictly speaking, Planck initially thought that the quantization
was only a function of the emission process and not inherent to the light itself.
1905 (Einstein): Albert Einstein stated that the quantization was in fact inherent to the
light, and that the lumps can be interpreted as particles, which we now call “photons.” This
proposal was a result of his work on the photoelectric effect, which deals with the absorption
of light and the emission of elections from a material.
We know from Chapter 8 that E = pc for a light wave. (This relation also follows from
Einstein’s 1905 work on relativity, where he showed that E = pc for any massless particle,
an example of which is a photon.) And we also know that ω = ck for a light wave. So
Planck’s E = h̄ω relation becomes
E = h̄ω =⇒ pc = h̄(ck) =⇒ p = h̄k (2)
This result relates the momentum of a photon to the wavenumber of the wave it is associated
with.
, 10.1. A BRIEF HISTORY 3
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 = h̄ω and p = h̄k. 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 h̄ 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 = h̄ω and p = h̄k 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 wave-
particle 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 am-
plitude. 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 un-
random. 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.
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