Download A Motivation of Quantum Mechanics

A Motivation of Quantum Mechanics
Robert C. Helling ([email protected])
I have received a number of questions from mathematics students in our “Mathematical Quantum Mechanics”
course to motivate the “axioms” of quantum mechanics that were given in the lecture. This text is my attempt
to partially fulfil this request. Of course, there are many different things that could be called “motivate
quantum mechanics” and this is only one: I will try to explain how a number of simple Gedankenexperimente
can lead one to study Schrödinger’s equation. Of course, as always, nothing will be proved and in fact, some
of my claims probably cannot be proven. But still, what I explain here is firmly believed to be true by most
physicists.
Of course, we will not be able to derive something (the formalism of quantum mechanics) from nothing.
We have to take some input. Therefore, I will not really explain quantum mechanics from scratch and pure
thought. Rather, I will tell you about the outcome of a number of hypothetical experiments and then use
intuition to suggest a formalisation of the experimental outcomes. This will not be a deduction in any strict
sense. Rather I claim, this interpretation and formalisation then leads to predictions for other experiements
which are in fact confirmed. So eventually, I will also have to withdraw to a “this is what works” type of
argument. However, for some people the things I have to claim hold true might be more natural or easier to
believe.
Once again, what I will not explain is a more well defined procedure of “quantisation”, how to arrive at some
Hilbert space and Hamilton operator given some classical system. I will only motivate why the Hamilton
operators that we deal with have a chance to describe some aspects of nature.
These notes follow in spirit the description by Feynman in the beginning of volume III of his lectures. You
should consult those if you want a better and longer explanation.
I will discuss here a number of incarnations of the “double slit experiment”. Of course, this these are only
thought experiments, we are not actually doing them. The outcome of the classical experiments, however,
should be intuitively clear whereas quantum mechanics makes specific predictions for the outcome of the
quantum experiment. In fact, in the last decade (that is much later than the time at which quantum
mechanics was developed as a theory) quantum optics experiments have been conducted that are at least in
spirit similar to the double slit experiments I will be discussing and that confirmed all the predictions based
on quantum mechanics.
All these thought experiments consist of a source of whatever we will be studying (a shot gun, an emitter
of water waves, an electron gun like in an old fashioned TV tube) then a wall with a number of holes that
can be opened or closed and finally a screen that records where one of the things that were emitted hit. For
each of these experiments I will show a histogram of how many things hit the screen where.
The first experiment is using a shot gun shooting against a wall with two holes that can be blocked. Obviously,
if no hole is open nothing gets through.
Next, we open the upper hole. Now, some of the shot goes through the upper hole. As there is some scatter
around the hole, the distribution at the screen is a little bulge close to the open hole. Let us call this
distribution s1 (x). (Fig. 2).
Similarly, if we open the other hole, there is a bulge close to the other hole (Fig. 3). This distribution we
denote by s2 (x).
Now we open both holes. The result is what everybody would expect: Since now the shot can go through
either hole the two previous distributions just add (Fig. 4). In formulæ: s12 (x) = s1 (x) + s2 (x).
After we gained some experience with shotguns we change the set-up. Now we put the whole apparatus in
water and have the source be some little mechanism that emits surface waves in the water. Of course, now,
the height of the water surface oscillates with time. But what we record at the screen is the intensity of
the waves arriving, that is the square of the amplitude. At first, the result is very similar to what we have
observed for the shotgun: With no hole open nothing gets through, and with only one of the holes opened
there is a maximum of intensity close to the hole. Let us denote these intensities by I1/2 (x).
1
blocked
hole
source
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Fig. 1: Shotgun with both holes blocked: Nothing gets through
hole
source
wall
histogram:
hole 1 open
screen
Fig. 2: Shotgun with one hole opened
source
hole
0110
1010
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10101010
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1010
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wall
0110
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historgam:
hole 2 open
screen
Fig. 3: Shotgun with the other hole opened
In the case of two open holes, however, something different happens: There is interference, at some places,
a hill of one partial wave always meets a valley of the other and the net intensity is much smaller. At other
places, hills meet hills and valleys meet valleys and we have “constructive interference”. The pattern is
shown in Fig. 5.
2
hole
source
hole
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histogram:
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screen
Fig. 4: Shotgun with both hole opened
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hole
source
hole
wall
histogram:
both holes open (waves)
screen
Fig. 5: Waves through two holes display an interference pattern
This is most easily understood as follows: The time evolution of a single wave is Re(A1/2 (x)eiωt ) with some
complex amplitude A1/2 (x). The intensity for this wave is given by I1/2 (x) = |A1/2 (x)eiωt |2 . But for waves,
it’s not the intensities that add but the (complex) amplitudes: A12 (x) = A1 (x) + A2 (x) and as total intensity
of the sum of the two partial waves we have I12 (x) = |A1 (x) + A2 (x)|2 6= I1 (x) + I2 (x).
Finally, we miniaturise the whole experiment and use electron as projectiles. As source, we use an electron
gun like the one used in cathode ray TV tubes. But we make sure that the rate of emission is so low that
at any time there is at most a single electron in the apparatus. When this electron hits the screen, there is
a little flash of light at the position where the electron hits. Hence each individual electron hits the screen
at a well defined position, just like in the case of the shotgun and unlike the case of the water waves which
reached the screen at all positions simultaneously.
Once more, we assemble a histogram of where how many electrons arrive at the screen. In the case where
only a single hole is opened the resulting histogram is similar to the experiments before. But when we open
both holes the distribution is no anymore like the one for the shotgun but in fact turns out to be like in the
case of the water waves (Fig. 5)! Still the individual electron arrive at specific points, it is just the counting
distribution that is like in the wave case.
Especially, there are places denoted by the valleys in the interference plot where fewer electrons arrive with
both holes open than with only one of the holes open! So, somehow, the electrons when they go through
one hole have to “know” that the other hole is open as well. Note well this is a single electron effect since
we made sure that there is only a single electron in the experiment at any time.
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From this, we have to conclude that the formalism must be like in the wave case: For each hole, there is
a complex quantity ψ1/2 (x) and the probability to measure the electron at position x is given by |ψ(x)|2 .
When both holes are open, rather than adding the probabilities (or rates) we first have to add the complex
quantities and only then take the absolute value squared: |ψ1 (x) + ψ2 (x)|2 6= |ψ1 (x)|2 + |ψ2 (x)|2 .
One could now have the idea that it would be interesting to detect through which hole the electron actually
passed. In order to do this, we put a little light close to hole number 1 in such a way that the electron would
reflect a bit of the light if it goes through that hole. However, we have to take into account that such an
observation can only be done if there is some interaction with the electron.
Once we change the experiment in this way, however also the distribution on the screen changes: A fraction
p of the electrons will be detected when going through hole 1 while the remaining fraction 1 − p of electrons
go through hole 2. If for our statistics we cound only the electrons passing through hole one, the resulting
distribution is as if we had closed hole 2, that is like in Fig. 2. Similarly, if we detect the electron going
through hole 2, the distribution is shown in Fig. 3. Obviously, if we count the electrons no matter through
which hole they went, the rates just add and we end up with the classical distribution Fig. 4. Just the presence
of the detector has changed the experiment so much that the interference pattern Fig.5 has disappeared.
How can this be understood? The interaction happens in such the way that we do not change the rate of
electrons going through that hole. Therefore, we must not change |ψ1 (x)|2 and thus we can only change ψ1
by a phase to ψ̃1 (x) = eiα ψ1 (x). The new intensity is then given by |ψ̃1 (x) + ψ2 (x)|2 . As we do not know
the phase α, we should express our ignorance by averaging over all possible values. Doing this, we find
Z 2π
1
dα |ψ̃1 (x) + ψ2 (x)|2
2π 0
Z 2π
1
=
dα |eiα ψ1 (x) + ψ2 (x)|2
2π 0
Z 2π
1
dα |ψ1 (x)|2 + e−iα ψ̄1 (x)ψ2 (x) + eiα ψ1 (x)ψ̄2 (x) + |ψ2 (x)|2
=
2π 0
= |ψ1 (x)|2 + |ψ2 (x)|2
And indeed, this is what we find in the experiment: As soon as we istall the detector which measures which
hole the electron took, the intensities |ψ1/2 (x)|2 add and not the the complex quantities ψ1/2 (x) and the
interference pattern goes away in favour of the addition of intensities as in Fig. 4!∗
This goes under the name of “decoherence” and is in fact the mechanism why the world that is quantum
appears to be classical (as with intensities adding). Here we have spelled out the mechanism in which an
interaction (in this case with the light) destroys phase information and thus the interference effects. Note
well that this is a qualitative effect: Instead of complex amplitudes we now have to add intensites.
This should not be confused with the classical limit h̄ → 0. The later is the limit of macroscopic systems.
Strictly speaking, h̄ is a dimensionful constant of nature and physically it makes no sense to take that limit.
Rather, h̄ is the scale at which the system (more precisely: variations of its action) are measured. So when
we say we take the h̄ → 0 limit we actually mean that we want to scale up our system holding h̄ fixed such
that the action measured in units of h̄ becomes large.
∗
You might be worried that this argument is not valid since nobody told us to weigh the different possibilities
α with equal weight in the integral. Here is a different, more general argument with the same result, although
using a somewhat more abstract formalism, namely density matrices which will be explained later in the
course: Effectively, the detector has two different orthogonal (otherwise the detector would not be effective)
states after the experiment: “The electron went through hole 1”, let us call that state |1i and “the electron
went through hole 2”, denoted |2i. The detection of the hole amounts to entangling the electron with the
detector, that is, after the hole has been detected by
√ some interaction of the electron with the detector the
whole system is in the state (ψ1 ⊗ |1i + ψ2 ⊗ |2i)/ 2. Observing hole 1 projects this state to just ψ1 ⊗ |1i
leading to Fig. 2 while detecting the hole but ignoring the outcome corresponds taking the partial trace
over the detector Hilbert space leading to a mixed state of the electron described by the density matrix
̺ = (|ψ1 ihψ1 | + |ψ2 ihψ2 |)/2.
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In principle, there is nothing that stops macroscopic systems from behaving quantum mechanical, from
showing interference effects etc. However, the larger the difference in action (measured in h̄) the larger
the effect of small perturbations (from interactions with the environment) on the phase. Thus, macroscopic
systems are much more susceptible to losing their phase information due to interaction with the environment
and to behave classically (that is without interference patterns but with addition of probabilities). Another
way to say this is to notice that the spacing of the interference pattern in Fig. 5 is related to h̄ and in
the small h̄ limit it becomes infinitely narrow and thus washed out. However an averaged out interference
pattern is nothing but the classical distribution Fig. 4. But if you had a very fine resolution you would still
detect the interference pattern no matter how small h̄ is.
What we have just seen is actually quite characteristic for quantum systems: You cannot make arbitrary
observations without changing the results of other observations. Here, it was impossible to study the origin of
the interference pattern by analysing which hole the electron went through: Once that observations is made,
the outcome of the interference pattern on the screen is destroyed! This does not just display our carelessness
with the experimental set-up, that maybe we have not been careful enough with our measurement. It turns
out that it is fundamentally built into quantum mechanics that any measurement which is sensitive enough
to detect which hole the electron went through at the same time destroys the interference pattern (actually,
we did not really specify how the measurement worked except that it multiplies ψ by an unknown phase).
Let us go back to the electron experiment without the light that gave us the interference pattern. It should
be obvious what would happen if instead of two holes our wall would have N : The total intensity would be
2
N
X
ψn (x) .
|ψ12···N (x)|2 = n=1
The second generalisation one can imagine is to have several walls with holes instead of just one. In that
case the ψ would have a multi-index indicating the sequence of holes in the different walls that the particle
took. Again, to obtain the rate at the point x on the screen, we would have to sum up the ψ’s for all possible
combinations and take the absolute value squared of the result.
The next step is to further increase the number of walls and the number of holes in each of the walls. Doing
this we end up with some sort of sieve between the source and the screen. We can go even further and
pretend to take a continuum limit of this procedure after which the whole space is filled by holes in walls
and in fact there are no walls remaining. To obtain the total ψ, we have to sum over all over all possibilities
to get to from the source to the screen, i.e. over all possible paths ~x(t):
ψ(x) =
X
ψ~x(t) (x)
~
x(t)
R
Physicists usually denote this sum over all possible paths by a “path integral” which they write as D[x(t)].
This has some relation to the Ito calculus (after a “Wick rotation”) but by itself should better be viewed as
some symbolic object rather than a well defined entity.
Beyond problems of properly defining the path integral we still have to figure out what the integrand is:
What is ψ~x(t) (x)? At this point, a certain amount of guesswork is required. I will use an analogy to argue
for one possibility but of course this is in no strict sense unique, and as I indicated above, this is what in
the end works.
To determine the contribution ψ~x(t) of an individual path, we first notice that we build our experiment
in such a way that there is only one possible path for example by installing a litle tube from the source
through a hole in the wall to the screen such that the only path the electron can take is through the hole,
the probablitly of the electron reaching the screen at the endpoint of the tube is 1. Thus, for a single path
ψ~x(t) should be a phase which of course depends on the path ~x(t).
Next, we have to make contact with the classical world. There is one positive parameter h̄ in quantum
mechanics that controls the “quantumness” of a system. Specifically, in the limit of h̄ → 0 we should recover
classical physics. In the language of the path integral, we require that in this limit, only those paths ~x(t)
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contribute that obey the classical equations of motion. For example we could obstruct all the other paths
that do not fulfill the classical equations of motion. In the limit h̄ → 0 this should not change the outcome
of the experiment.
A very convenient way to write down the classical equations of motion is in terms of a variational principle:
The “action” S of a path is defined to be the time integral over the Lagrangian L
S=
Z
dt L[~x]
where I used square brackets to denote a functional, that is L at time t not only depends on the position ~x(t)
but also the velocity ~x˙ (t) via the kinetic energy. The equations of motion can now be written as δS/δ~x(t) = 0
meaning that S is extremal upon variations of the path ~x(t). [I hope this is not news for you. If it is please
consult a textbook on analytical mechanics].
Our task now is to guess an expression for the complex phase ψ~x(t) (x) depending on h̄ in such a way that in
the limit h̄ → 0 only those ~x(t) contribute to the path integral for which S is stationary.
At this point, one might be reminded of the method of stationary phase† . There, one is interested the
asymptotic value of integrals of the form
I(Λ) =
Z
ds A(s)eiΛφ(s)
in the limit Λ → ∞ for sufficiently nice functions A and φ. It turns out, the leading contribution to these
integrals comes from those sk where φ is stationary, that is φ′ (sk ) = 0. One can show
I(Λ) =
X
k
s
2πi
A(sk ) eiΛφ(sk ) (1 + O(Λ−1 )).
Λφ′′ (sk )
Some care has to be taken to select the proper branch cut for the square root of a complex number but this
can be done.
The analogy should be clear: 1/h̄ plays the role of Λ and instead of a one dimensional integral over s we
have an infinite dimensional integral over all paths ~x(t) that in the classical limit only receives contributions
from stationary paths if let S[~x(t)] play the role of φ(s).
Obviously, this is by far not unique. For example, we could add to S other terms that vanish fast enough in
the limit h̄ → 0. However, such ambiguities are also inherent in the more familiar “canonical quantisation”
where upon replacing the position variable x by a multiplication operator and the momentum variable p
by −ih̄d/dx because of factor ordering: Classically xp = px but as operators there is a difference of ih̄. In
addition we could have a non-trivial amplitude corresponding to the A(x) in the one dimensional integral
which would be independent of h̄. However, one could appeal to the Hamilton-Jacobi form of classical
mechanics to argue that this is in fact not there.
By these arguments we are finally led to guess the expression of an integral over all paths ~x(t) that lead to
the point x
Z
S[~
x(t)]
ψ(x) = D[~x(t)] ei h̄
such that |ψ(x)|2 gives the rate at which electrons arrive at the point x on the screen.
There is now one calculation that we can do with this ψ(x) which is given in terms of this very ill-defined
infinite dimensional interval: One can compute how ψ(x) changes when we are not interested in electrons
arriving at time t but only at an infinitesimal later time t + dt, that is if the electron as slightly more time
to reach the screen.
†
see http://en.wikipedia.org/wiki/Stationary phase approximation
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I will not show this calculation here, but it is quite straight forward (actually, you should try to do it!): First
you realise, that since the path integral factors you only need to consider infinitesimal paths from time t to
time t + dt for those you can assume that the path is just a single straight step ξ. Thus you only have to
do a one dimensional integration over ξ instead of the path integral. Then you express L as mẋ2 /2 − V (x),
expand ψ(x − ξ) in a Taylor expansion to the quadratic order and do the Gaussian integral over ξ (after
an appropriate change of variables). Finally, you expand the exponential and collect terms and you find a
partial differential equation that ψ(x) obeys.
As it turns out, this differential equation is the time dependent Schrödinger equation
ih̄
∂
ψ = Hψ
∂t
for the Hamilton operator H that you obtain from L by first doing a Legendre transformation to a Hamilton
function and the replace x and p according to the rules of canonical quantisation.
So, to sum up, if you believe my interpretation of the double slit experiments and the generalisation to path
integrals then you obtain a wave function that obeys Schrödinger’s equation. If you want this is a derivation
of Schrödinger’s equation from the path integral (which suggested itself from an analysis of double slit
experiments).
However, this is not how the subject evolved historically. Feynman came up with the path integral argument
in the 1940s in his PhD thesis whereas Schrödinger published his equation already in 1926.
The founding fathers of quantum mechanics had to explain a number of experiments which were in conflict
with classical physics. One of them was that atoms emit light only at discrete frequencies and not with a
continuous spectrum. From experience with ordinary and partial differential equations gained in th 19th
century, those people knew that for some differential operator D equations of the type Df = λf only had
solutions for special values of λ which suggests that if one manages to relate the frequencies to λ one might
be able to find a criterion for the allowed frequencies.
From the “Hamilton-Jacobi-formalism” for classical mechanics, Schrödinger was able to derive a differential
equation of the above type that was able to reproduce the allowed frequencies for the hydrogen atom. But
the quantisation procedure that led from the classical system to the quantum operator was not justified at
that time.
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