The Heisenberg Uncertainty Principle is a very important and central part of quantum mechanics. It is one of the primary features that distinguishes quantum and classical mechanics and gives rise to "quantum weirdness". The most familiar form is the original
position-momentum uncertainty relation, which can be stated as follows:

Δx Δp ≥ hbar/2

where Δx is the uncertainty in position, Δp is the uncertainty in momentum,
and hbar is Dirac's constant, which is Planck's constant divided by 2π. hbar is
small, but it is not zero, which tells you that for a minimum uncertainty state^{1} Δx and Δp are inversely proportional. That means that if one is small the other must be large.

Those are the mathematical facts behind the basic relation, but understanding what it actually means takes some more work. The uncertainty principle is a frequent subject of modern physics abuse syndrome and is often even misunderstood by scientists. Below I will attempt to explain a little about what it means and what it *does not* mean. The position-momentum uncertainty relation is actually part of a more general mathematical relation in quantum mechanics, which is sometimes called the generalized uncertainty principle (to distinguish it from the specific one for position and momentum). There is also an energy-time uncertainty principle that is superficially similar but is not actually an example of the generalized uncertainty principle. First I will give an informal explanation of the uncertainty principle, then a more formal explanation and a proof. I will look at the topic from the standpoint of our modern understanding (and use roughly the Copenhagen Interpretation for those experts in the audience) not necessarily how early quantum physicists
like Bohr and Heisenberg understood it.

Quantum mechanics usually doesn't tell us exactly what outcome a measurement performed on a physical system will have, it only tells us which outcomes are possible and
how likely each outcome is. Even if you have
two systems that are in identically the same state, quantum mechanics says you can get different results from making the same measurement on each system. For example, quantum theory tells you the probability of finding the electron in a hydrogen atom within a given distance of the
nucleus, though it doesn't tell you where the electron will be specifically. If you were to take two hydrogen atoms in the ground state and measure the position of the electron in each atom, you would in generally get different results. That is to say that the outcome of measurements in quantum mechanics is nondeterministic.

If you have such a probability distribution for an observable quantity O, you can define an expectation value, denoted ⟨O⟩, which is the value you'd get if you averaged the results of measurements made on many systems all in that quantum state. Since a range of different possible outcomes happen, there is a sort of natural spread of values around the expectation value that are still fairly likely to occur for any one measurement. This characteristic spread can be
measured mathematically by finding the standard deviation of the set of possible outcomes, and that gives us the uncertainty in the value of O, which we'll call σO. So the uncertainty in O is the spread of possible values that might result from measuring O, and it gives us an idea of roughly how far we might expect any one measurement to differ from the average value of many measurements done on identical systems.

#### The Meaning of the Position-Momentum Uncertainty Relation

Now that we have an idea of what "uncertainty" means, we can return to the position-momentum uncertainty relation. The relation

Δx Δp ≥ hbar/2

means the following: Suppose that you had a very large number of identically prepared quantum systems^{2}. On
half of the systems you measured the position, x, and on the other half you measured the momentum, p. Then you used the standard deviation to find the uncertainty in x (from your x measurements) and the uncertainty in p (from your p measurements). The uncertainty principle is saying that those two uncertainties would have the relationship given above. This is true for any possible quantum state. If you choose to put a system in a
state where it has a very well defined position (very small spread of possible positions), then it will have
a very poorly defined momentum (a very wide range of values). This is a statement about the fundamental
nature of the quantum state. As Heisenberg puts it, "This indeterminateness is to be considered an essential characteristic of the electron, and not as evidence of the inapplicability of the wave picture."^{3}

#### Implications and Misconceptions

Probably the most common misconception is that the Heisenberg Uncertainty Principle is equivalent
to the statement, "You can't measure a system without changing it." If you look at the way I explained
it above, the position and momentum measurements are being done on entirely separate systems, which
happen to have started out in the same state, so it is not caused by the first measurement interfering with a second. In fact, we haven't said anything about what state the systems are in after a measurement.

Now, in quantum physics when you measure an observable of a system, the system undergoes what is sometimes
called a wavefunction collapse^{4} so that afterward it is in a new quantum state, and with most sorts of measurements if the same measurement is repeated immediately you will always get the same result a second time. That means if you measure the
position of that electron in the hydrogen atom to high precision and find it to be at a specific place,
then if you measure the position again immediately you'll find it in the same place. Thus, after the first
measurement the electron must have collapsed into a quantum state with a well defined position (even though
it did not have one before the measurement). Then the uncertainty principle tells us that this new state must
have a poorly defined momentum (a wide spread of possible values), so if we did try to measure the momentum
of the electron after we measured its position, we'd just get some really uncertain nonsense (a wide and erratic spread of values). As a result
we could not get any information about what the momentum originally was from those later momentum measurements.
So it is true that a measurement of a particle's position will destroy any information about it's momentum
(and vice versa), but the point is that this is a result of the uncertainty principle together with the
entirely separate assumption of wavefunction collapse. Since the uncertainty relation doesn't depend on making measurements on the same system, only identically prepared systems, the uncertainty principle must be a statement about the inherent uncertainty in the original state.

Many people think that the uncertainty principle has to do with measuring devices just being flawed and not
being able to make precise measurements, but notice that the uncertainty principle allows us to have Δx to
be as small as we like, as long as we don't care about Δp. Or we can have Δp as small as we like, as long as we don't care that Δx is large. Beyond that, notice that I've been insisting that the uncertainty
depends on the quantum state of the system and haven't mentioned anything about the measuring apparatus. In
fact, the uncertainty principle we've been discussing assumes the measuring apparatus can make a perfectly
precise measurement. The uncertainty is in the intrinsic "fuzziness" of the system. We are not even taking
into account actual measuring apparatus that have other sources of imprecision.

One often reads or sees "derivations" of the uncertainty principle from ideas like trying to observe atoms with a microscope and considering the recoil from each photon hitting the atom. When quantum mechanics was first developed, people often spoke of this idea and, indeed, Heisenberg discusses it^{5}. This is seen by modern physicists as what might be called a "motivation": it motivates one to consider certain ideas but it is not actually a well formed proof.
It seems you might reasonably think that perhaps a particle really *does* have both an exact position and momentum, and we
just can't measure what they are. That sort of theory falls under a class of ideas called
hidden variables theories, and it turns out that Bell's Theorem tells us that, under
fairly general assumptions, there should be a testable difference between those sorts of theories and quantum
theories. The details of that proof are beyond the scope of this writeup. Subsequent experiments suggest that the physical world agrees with quantum
mechanics, so it appears that it's not just that we can't measure the position of the electron in a hydrogen atom precisely, it's that
**the electron really doesn't ***have* a precisely defined position! That is weird in the
extreme, but as far as we can tell it seems to be the truth. This also means that the uncertainty principle cannot be derived from classical mechanics, and while you can motivate why you might consider such an idea, you can't really "make sense of it" from classical mechanics alone.

In fact, once you know de Broglie's relation and the statistical interpretation the momentum-position uncertainty relation comes fairly directly from the mathematical theory of waves. See the de Broglie's relation node for more details.

#### Where is the Uncertainty Principle in Everyday Life?

If the uncertainty relation is supposedly so fundamental, why didn't anyone notice it was there until the 20th century? Why can't you observe it when you're bowling or when two cars crash? This question relates to the fairly profound issue of correspondence between quantum and classical mechanics, which includes the interpretation of quantum mechanics and the study of quantum decoherence, but we can give a pretty good, simple answer for every day situations. The answer? **hbar is small**. One way you can often get the "classical limit" in quantum mechanics is to set hbar = 0. In the case of the position-momentum uncertainty principle, that means Δx Δ p ≥ 0, which is by definition true even in classical physics, since both of the uncertainties are positive numbers. Now, of course, in real life hbar is not zero, but it turns out that it's pretty small. hbar is approximately 10^{-34} J s. This means that is you have a 0.5 kg ball and measure its position to an precision of 1 μm (a fraction the width of a human hair), the uncertainty principle implies that the uncertainty in velocity must be at least 10^{-28} m/s. In everyday life there's no way you can tell the velocity to that sort of precision. To give some clue of how precise this is, consider that at 10^{-28} m/s it would take an object more than 300 billion years for that object to move 1 nm (one millionth of a millimeter). So, the bottom line is that the minimum uncertainties imposed by the uncertainty principle are way too small to notice in everyday situations. You wouldn't expect to see them until you start doing *very* precise measurements or deal with very small systems.

#### The Energy-Time Uncertainty Principle

Often times people quote an energy-time uncertainty principle, but it is not the same as the
one we've been discussing, the one used for position and momentum. Although that statement is usually given in a similar mathematical form, it's different because time is not an observable in quantum mechanics, as position and momentum are. This should not be too surprising, since you don't measure a system to find its time like you would to find, say, its momentum. Rather, you think of time as existing independent of the system, as something
that the observer keeps track of. For this reason there's some ambiguity about what you might mean by "uncertainty in time", and the same mathematical arguments can not be used. If you define what you mean by "uncertainty in time" in a clever way, though, you can get a relation that looks superficially the same as the position-momentum uncertainty relation.

### The Generalized Uncertainty Relation

The position-momentum uncertainty relation is actually just a special case of a more general theorem in quantum mechanics sometimes called the generalized uncertainty principle. The term "Heisenberg uncertainty principle" is used to refer either to the general principle or the position-momentum uncertainty principle, though the latter was the one Heisenberg himself originally stated. The generalized uncertainty principle says that, in quantum mechanics, if you have two
observable quantities of a system, then there will in general be some lower bound to the
uncertainty with which both values can be known for both observables. Mathematically, for
two observables A and B, ΔA ΔB ≥ L. In general, this lower bound is not zero, meaning
that the less the uncertainty in one the greater the uncertainty in the other. The lower
limit is defined by the commutator of the operators representing the two observables, denoted
[A,B], which roughly measures how similarly each acts on the
specific system. So in general L is dependent on both of the observables being
discussed and the state of the system. This is defined for any pair of observables, which
can include things like position, momentum, energy, orbital angular momentum, and spin.

The form of the Heisenberg Uncertainty Principle that is normally discussed is the
statement of it for position and momentum. For position x and momentum p the lower
bound on uncertainty turns out to be independent of the state, and results in the uncertainty
relation

Δx Δp ≥ hbar/2

In fact, there are many pairs of observables that obey exactly the same uncertainty relation. Two such observables are said to be "canonically conjugate" to one another, which is a term from classical mechanics (specifically Hamiltonian mechanics). Some examples of canonically conjugate pairs of observable are the following: position and momentum, the components of angular momentum along two perpendicular axes, and a component of linear polarization and a component of circular polarization of light. The generalized uncertainty principle is a very nice and useful mathematical result, but it is general enough that only so much can be said without resulting to a lot of math, which is what I will do now.

For an observable O, let ⟨O⟩ be the expectation value of
O and let ΔO be the standard deviation of O, so
(ΔO)^{2} = ⟨O^{2}⟩ - ⟨O⟩^{2}. For a complex number z, let
conj(z) be the complex conjugate of z and |z|^{2} be the complex norm squared, z*conj(z). Let
adjoint(O) be the adjoint of O, also called the hermitian conjugate,
and remember that all operators in this proof represent observables, so they are all
hermitian (self - adjoint). Let the commutator of A and B be
[A,B]=AB - BA.
Finally, hbar is Planck's constant divided by twice pi.

Given two observable
quantities represented by operators A and B:

(ΔA)^{2} (ΔB)^{2} ≥ |⟨[A,B]⟩|^{2}/4

This is the general statement of the Heisenberg Uncertainty Principle for any two
quantities you can observe. The value of the lower limit of the uncertainty depends on
what you're measuring, and it can be zero in some cases. This is not the same as the
energy-time uncertainty relation because this is stated for observables of the system as
represented by operators on the Hilbert space of the system. Time is not an observable,
and it is represented in quantum mechanics as a scalar parameter of the theory, an
independent variable, not an operator. Also, the sense in which uncertainty is defined is
different. There is extensive discussion of the meaning of these relations above, but it is worth pointing out that this theorem applies to the situation where observable A is measured on one group of systems and observable B is measured on an entirely different groups of systems as long as they are prepared in the same quantum state.

For this proof, I will use Dirac notation, where a state of the system labeled ψ, a
vector in the Hilbert space of the system, is denoted by the ket |ψ⟩ and its
dual is denoted by the bra ⟨ψ|. The
inner product of a bra φ with a ket ψ is then denoted ⟨φ|ψ⟩.
Also, the expectation value of an observable O for a
state ψ is ⟨O⟩ = ⟨ψ|O|ψ⟩, where O is a
hermitian operator on the Hilbert space.

Let the system be in a state |ψ⟩, and consider two observables represented by the
operators A and B. To prove the theorem as stated above, we will need to restate both sides of the
inequality.

First, consider an operator δA, the gives the deviation of A from the average value.

δA = A - ⟨A⟩

⟨δA^{2}⟩ = ⟨(A - ⟨A⟩)^{2}⟩ =
⟨A^{2} - 2 ⟨A⟩ A + ⟨A⟩^{2}⟩ =
⟨A^{2}⟩ - 2 ⟨A⟩ ⟨A⟩ + ⟨A⟩^{2}

Thus, ⟨δA^{2}⟩ = ⟨A^{2}⟩ - ⟨A⟩^{2} = (ΔA)^{2}

And the same goes for B and δB.

The next we need to work out some facts about these deviation operators, δA and δB. We can fairly easily determine they have the same commutation relations as A and B.

[δA,δB] = [A - ⟨A⟩,B - ⟨B⟩] =
[A,B] - [⟨A⟩,B] - [A,⟨B⟩] + [⟨A⟩,⟨B⟩]

⟨A⟩ and ⟨B⟩ are just numbers so they commute with any other object; thus,

[δA,δB] = [A,B]

Also, both δA and δB are hermitian, since the
Adjoint(A - ⟨A⟩) = Adjoint(A) - conj(⟨A⟩) = A - ⟨A⟩
given that A is hermitian (which also implies that ⟨A⟩ must be a real number).

Now we can proceed with the meat of the proof.

⟨[δA,δB]⟩ = ⟨δAδB - δBδA⟩ = ⟨δAδB⟩ - ⟨δBδA⟩

⟨δBδA⟩ = conj(⟨adjoint(δA) adjoint(δB)⟩) = conj(⟨δAδB⟩)

since δA and δB are hermitian. Thus,

⟨[δA,δB]⟩ = ⟨δAδB⟩ - Conj(⟨δAδB⟩) =
2 i Im(⟨δAδB⟩)

where Im(z) is the imaginary part of z. Now obviously

|⟨[δA,δB]⟩|^{2} = 4 |Im(⟨δAδB⟩)|^{2} ≤
4 |⟨δAδB⟩|^{2}

Since the norm of the imaginary part can't be more than the norm of the entire complex number. Now we can pull a trick by writing

|⟨δAδB⟩|^{2} = |⟨ψ|δAδB|ψ⟩|^{2}

and considering that as the inner product of two states |ψ_{A}⟩ = δA |ψ⟩ and |ψ_{B}⟩ = δB |ψ⟩. Then by the Cauchy - Schwartz inequality, for any two vectors |φ⟩ and |ψ⟩,

|⟨φ|ψ⟩|^{2} ≤ |⟨φ|φ⟩| |⟨ψ|ψ⟩|, so

|(⟨ψ|δA)(δB|ψ⟩)|^{2} = |⟨ψ_{A}|ψ_{B}⟩|^{2} ≤|⟨ψ_{A}|ψ_{A}⟩| |⟨ψ_{B}|ψ_{B}⟩| =
|⟨ψ|δAδA|ψ⟩| |⟨ψ|δBδB|ψ⟩| =
|⟨δA^{2}⟩| |⟨δB^{2}⟩| =
⟨δA^{2}⟩ ⟨δB^{2}⟩

since ⟨δA^{2}⟩ and ⟨δB^{2}⟩ have to be positive because δA and δB are hermitian.

Now to put all this crazy crap together:

(ΔA)^{2}(ΔB)^{2} = ⟨δA^{2}⟩ ⟨δB^{2}⟩
≥ |⟨δAδB⟩|^{2} ≥
|⟨[δA,δB]⟩|^{2}/4 = |⟨[A,B]⟩|^{2}/4

So there it is. The proof itself is perhaps not too enlightening, which is that why I saved it until last, but now you know. Of course, this is a modern proof based upon formalism developed after Heisenberg stated
the uncertainty principle. His proofs were based on similar principles but he was not thinking of things in terms of Hilbert spaces, and original proofs actually didn't come up with the correct minimum limit, as he had Δx Δ p ≥ hbar.

This was my very first node. I largely rewrote it (as of 11/04) and the original now resides at the bottom of my homenode. Let me know if you think this is an improvement. Any suggestions on how the node can be clearer (especially for people who don't know much about the subject) would be very welcomed.

Notes

- A "minimum uncertainty state" is one in which the product Δx and Δp is minimized, meaning that if you choose a value for one, then this state has the minimum possible for the other.
- A system is usually prepared in a certain quantum state through a series of operations and measurements of the system. For example, an electron may be prepared in a spin up state by sending it through a Stern-Gerlach device (which separates out to beams of electrons, one with spin up and one with spin down) and using only electrons from the spin up beam.
- Heisenberg, p. 14.
- Unfortunately, the term "wavefunction collapse" can actually have a variety of meanings to experts. Here I just mean a projection onto an eigenstate of the measurement not making any statement about a dynamical process by which the state undergoes this projection. This is what von Neumann referred to as "process 1".
- Heisenberg, p21.

Sources

- The Physical Principles of the Quantum Theory, Werner Heisenberg
- Introduction to Quantum Mechanics, David J. Griffiths
- Modern Quantum Mechanics, J. J. Sakurai