Download Notes for Class Meeting 19: Uncertainty

Notes for Class Meeting 19: Uncertainty
Uncertainty in Momentum and Position
In 1926, Werner Heisenberg formulated the uncertainty principle: It is impossible to
determine both the momentum and position of an object to infinite precision. Explicitly,
!
!x!px " ,
(19.1)
2
or in words, the uncertainty of the position of an object in some direction times the
uncertainty in its momentum in that direction is always greater than or equal to ! / 2 .
A couple of points should be noted:
(1) It is only the product of the uncertainty of the position in some direction and the
uncertainty in the momentum in that direction that enters the uncertainty principle. For
example, there need not be any inherent uncertainty between the position in the x
direction and the momentum in the y direction.
(2) The value ! / 2 is the absolute minimum uncertainty and only occurs in one
special case. More common values are ! or h/2, depending on the case, but the exact
value will not be important for our purposes.
In his lecture, Feynman explained that the uncertainty principle arises because any
attempt to measure the position of a particle will disrupt its momentum. This is not the
best explanation, but let’s explore it first. Why is the two-hole experiment an example of
the uncertainty principle?
Let’s solve for the first minimum in the two-hole interference pattern, which we will
call x, the distance from the center maximum. This will occur when the distance from
one hole is a half wavelength longer than the distance from the other hole, since this will
cause the peak of a wave from the first hole to coincide with the trough of a wave from
the other hole. Thus, the condition is
(19.2)
L2 ! L1 = " / 2 .
From the Pythagorean theorem,
2
2
"d
%
"d
%
L = L + $ ! x ' and L22 = L2 + $ + x ' .
(19.3)
#2
&
#2
&
After a bit of algebra, which we will consign to the appendix, we obtain
x
!
.
(19.4)
=
L 2d
The right hand side of Eq. 19.4, x/L is the change in angle, !" , between the center
maximum and the first minimum of the interference pattern. Thus, the change in the x
component of the electron momentum, px , that would destroy the interference pattern is
just
h
h #
h
,
(19.5)
! px = p !" = !" =
=
#
# 2d 2d
where we have used the de Broglie relationship that p = h / ! . Therefore, to avoid
destroying the interference pattern, we must change the x component of the electron
momentum by less than h/2d.
2
1
2
But to detect which hole the electron went through, we need to use a photon with a
wavelength less that the distance between the two holes,
(19.6)
!<d,
This is because, in some sense, the photon size is comparable to its wavelength.
From the de Broglie relationship, the photon has a momentum
p! = h / " .
(19.7)
Bouncing a photon of this momentum off of the electron would cause a change in the x
component of the electron momentum by about
h
h
(19.8)
!px " 2 p# = 2 > 2 ,
$
d
where the last piece of Eq. 19.8 comes from using Eq. 19.6. Thus, since the change in the
x component of the electron’s momentum, Eq. 19.8, is larger than that required to destroy
the interference pattern, Eq. 19.5, the interference pattern will be destroyed by detecting
which hole the electron went through.1
The above argument, which Feynman made in his lecture, has two advantages:
(1) A single counterexample can show that quantum mechanics is inconsistent (which
is what Einstein tried to do in the Einstein-Bohr exchanges).
(2) It uses classical arguments, so it is easy to understand.
However, I stated above that this argument is not the best explanation for the uncertainty
principle, and the reason is that it has two serious disadvantages:
(1) You can never list all of the ways of detecting which hole the electron went
through, so you cannot use it to establish a general principle.
1
A more precise analysis involves using a lens to focus the photon, but this is not important for our
purposes. In fact, it is important for you to understand the physics argument that is being made, but not
important for you to follow all of the mathematics.
(2) More important, it encourages you to think that the electron has a position and
momentum, but that you simply cannot measure it. This is incorrect. At any given time,
the electron has neither a definite position nor a definite momentum.
The reason why Feynman is sure that the uncertainty principle will hold in every
case, even though he cannot examine every case, is that the uncertainty principle is
inherent in the mathematics of waves. There is a mathematical theorem known as the
Fourier theorem, which states that any curve can be made from a (possibly infinite) sum
of sine and cosine functions of different wavelengths. A single sine function, say
sin(2! x / " ) , has a single wavelength ! and thus corresponds, through the de Broglie
relationship, to a single and absolutely precise momentum. However, it extends over all
space. If you want to localize the wave, you have to combine several different
wavelengths, and thus have some uncertainty in its momentum. The more the wave is
localized in space, the more uncertain the momentum becomes. I will demonstrate this
during the class meeting.
We will discuss the concept of measurement in quantum mechanics in the next set of
notes, but we will just note here that if one has a particle with a range of position and
momentum then a precise measurement of either will change the wave to make the other
more uncertain:
What the precise value of x or p becomes is purely probabilistic. This is the part of
quantum mechanics that, according to Feynman, “nobody understands.”
The uncertainty principle guarantees that the future is unpredictable in principle, as
well as in practice. If you know where a particle is precisely, then you do not know
where it is going. If you know where it is going (i.e., its velocity from its momentum),
then you do not know where it started.
Why do we not see the consequences of the uncertainty principle in our everyday
lives? Simply because h is extremely small, so the uncertainties are not noticeable in
macroscopic objects.2
Uncertainty in Energy and Time
What does the uncertainty principle tell us about time in quantum mechanics?
Consider a particle moving in the x direction. Energy and momentum are related,
1 2
1
p2
2
E = mv =
(mv) =
,
(19.9)
2
2m
2m
so any uncertainty in momentum will also mean an uncertainty in energy. With a bit of
calculus, one can show that
p
mv
1
(19.10)
!E = !p =
!p = v!p " !p = !E .
m
m
v
If we sit at one point in space and ask at what time does a particle pass us, then an
uncertainty in x will imply an uncertainty in the time of passage:
If “time early” is t = 0, then “time late” is t =
2
x
.
v
This is not completely true. A favorite problem for beginning quantum mechanics students is to ask them
to calculate how long you can balance a perfectly sharpened pencil on its tip. The uncertainly principle will
require the eraser end of the pencil to have some uncertainty in both its position and its momentum. This
will cause it to fall over in less than four seconds.
Therefore, !t =
1
!x " !x = v!t. Combining this with Eq. 19.10, we obtain
v
!
"1 %
!px !x = $ !E ' ( v !t ) = !E !t (
(19.11)
#v
&
2
What are we to make of Eq. 19.11? In standard textbooks on quantum mechanics,
you will find statements that time is just a parameter and cannot have any uncertainty.
The !E !t " ! / 2 relationship is interpreted as follows: An energy can only be measured
to a certain precision in a finite amount of time. 3
While I certainly agree with the above statement, I want to assert a more
unconventional interpretation of !E !t " ! / 2 , since I believe that you are already too
sophisticated to accept the standard interpretation as the full interpretation. We have
learned through relativity that space and time are closely related and form a fourdimensional spacetime. Therefore, if a particle’s position is uncertain, then a particle’s
time must also be uncertain. Thus, it follows that time in quantum mechanics is fuzzy in
the same way as space is fuzzy.
Consequences of the Fuzziness of Time
This concept can help us understand why energy states are quantized in the Bohr
atom. Say we want to measure the energy of an atom to high precision. From the
uncertainty principle, this will take some period of time. Consider two cases, one that
meets the Bohr condition that each orbit contains an integer number of wavelengths and
one that does not.
If a measurement takes some period of time, it implies that we are averaging over that
time. Assume that the Bohr condition is not met. Then on each revolution, the wave will
have a different phase, and over a long period of time the peaks and troughs will all
average to zero. However, if the Bohr condition is met, then all of the waves from
successive revolutions will be in phase with peaks matching peaks and troughs matching
troughs and we will have constructive interference.
This picture then explains why only discrete orbits exist in the Bohr model. It also
goes further in explaining why excited states, which decay by emitting a photon, do not
have a completely precise energy, but in fact have some spread in energy given by the
!E !t " ! / 2 uncertainty principle. If a state lives for a short time, then if the energy is
not quite at the center value, it will not have time to get completely out of phase before it
decays, since the peaks will move only a small amount on each successive revolution.
To be concrete, assume that a state lasts for 100 revolutions, then the energy (which
determines the wavelength) will have to agree with the Bohr condition to about 1 part in
3
I draw your attention to the sourcebook reading on the Bohr-Einstein exchange in which Einstein tried to
give a counterexample to this uncertainty principle. See also my interpretation of this in problem 246 in the
sourcebook (p. 306, with the solution on p. 385).
100. If we measure the energy of the photon, it will vary from event to event by about
1%. If there is only time for about one revolution, then any orbit is as good as another
and there is no energy quantization.
Comparison of the Position-Momentum and Time-Energy Uncertainty Principles
Even though we derived the time-energy uncertainty principle from the positionmomentum uncertainty principle, they are quite different in general due to the vector
character of momentum. For example, consider a hydrogen atom in its ground state. The
position-momentum uncertainty relates the uncertainty in momentum, which is about
twice its magnitude, since we do not know which direction it is in, to the diameter of the
atom, since we do not know where the electron is in the atom. However, the time-energy
uncertainty for the ground state is that the energy can be perfectly defined since the
ground state can last forever.
Quantum Tunneling
Another consequence of the fuzziness of time, or if you prefer, the time-energy
uncertainty principle, is the quantum mechanical property of tunneling. Consider a
particle trapped in a potential well, for example a ball rolling back and forth on a track, as
shown below on the left. If the dashed line shows the total kinetic and potential energy of
the ball, then it can never escape, because it does not have enough energy. However, the
time-energy uncertainty relation tells us that energy conservation can be violated, if it
happens for a sufficiently short time. Thus, a “quantum mechanical ball” would have
some probability of escaping the potential well and appearing on the other side, as shown
on the right.
Energy
Energy
Position
Position
Quantum tunneling is an important effect. It is responsible for particle decays, some
effects in electronics, and the burning of hydrogen in the sun. We will demonstrate a
form of quantum tunneling in the class meeting.
Appendix: Algebra to Get from Eq. 19.2 and 19.3 to 19.4
To recap, the condition for destructive interference is
L2 ! L1 = " / 2 .
(19.2)
From the Pythagorean theorem,
2
2
"d
%
"d
%
L21 = L2 + $ ! x ' and L22 = L2 + $ + x ' .
(19.3)
#2
&
#2
&
Subtracting the two equations in Eq. 19.3 from each other,
2
2
"d
%
"d
%
2
2
2
2
L2 ! L1 = L + $ + x ' ! L ! $ ! x ' (
(19.12)
#2
&
#2
&
d2
d2
2
(L2 ! L1 )(L2 + L1 ) =
+ dx + x !
+ dx ! x 2 = 2dx.
(19.13)
4
4
Since L ! d , it is a good approximation to set (L2 + L1 ) ! 2L in the left hand side of Eq.
19.13. We also use Eq. 19.2 for (L2 ! L2 ) to obtain
!
(19.14)
2L = 2dx "
2
x
!
.
(19.4)
=
L 2d