In quantum mechanics, observables correspond to Hermitian operators on the quantum state . This state is a vector in a complex Hilbert space; we have the inner product between two vectors, which is in bra-ket notation.

Now, for a Hermitian operator corresponding to some observable quantity, the expectation value of that observable for a quantum state is .

Now, remember that for a Hilbert space, as for all inner product spaces, we have the norm . We also note that for any two Hermitian operators and and any state vector , we have

,

where the bar denotes the complex conjugate. These arise because the operators are Hermitian.

For two operators *A* and *B*, the Cauchy Schwarz inequality tells us

Now,

,

where is the commutator of *A* and *B*.

However, we note

,

and similarly for *B*. Thus, our above inequality can be written:

Now, we recall in probability theory that the variance of a random variable X is

, and the standard deviation is the square root of the variance. Now, for two operators *A* and *B*, let us define and .

We see that as and are numbers,

Suppose we define for *A* a quantity *ΔA* analogous to the standard deviation for a random variable:

. Let us also analogously define *ΔB*. Then we see that

.

Using the inequality we have derived, we see that this means

, or, since ,

.

This inequality is, in fact, the general form of the Heisenberg Uncertainty Principle; it tells us the uncertainty relation between any two observables whose corresponding operators do not commute. For example, given the (one-dimensional) position operator *X* and corresponding momentum operator *P*, we have , so for a valid (normalized) state vector, giving us the usual form of the Uncertainty Principle:

.

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November 7, 2008 at 1:00 am |

[…] Physics Friday 45 Quantum Mechanics and Momentum Part 2: Wavefunctions, Operators, and Observables In quantum mechanics, the state of particle or system is represented by a wavefunction, which is a complex-valued function over some space. In more particular, methematical terms, the state of a quantum system is a vector in some complex Hilbert space. Usually, we represent the wavefunction as a function over some space, usually our standard position space. For one dimension, we have ψ(x). The probability density is given by the squared norm of the wavefunction; the probability Pab of finding the particle’s position in the interval (a,b) is . We see that to have a valid wavefunction, the probability for all of the space must be unity: ; such a wavefunction is said to be normalized. We have a similar situation for spaces of higher dimensionality; with the n-dimensional position vector, the probability of the particle being in a region V is , and normalization requires that . Note that the space over which the function is defined need not be physical space. For example, one can define a wavefunction ψ(p) over momentum space. In some situations, the wavefunction can be a vector of countably-infinite dimension, or even a vector of finite dimension; the wave function for a spin 1/2 particle (ignoring spatial freedom) can be represented as a two-dimensional complex vector (see here). Of particular importance is how we treat observable quantities of a system. Each observable property corresponds to a linear operator on the wavefunction, whose eigenvalues are the allowed values of the observable, with the corresponding eigenfunction for each value being the wavefunction for the state where the observable has that value. In other words, if we have an observable corresponding to the operator Â, then the observable has value a when the wavefunction for the system is ψa, where Âψa=aψa. In particular, the operators corresponding to physical quantities are hermitian. This, amongst other things, ensures that the eigenvalues are real. (Note that for an operator with discrete eigenvalues, eigenvectors corresponding to different eigenvalues are orthogonal in our vector space. Observables with a continuum of allowed values, however, give rise to eigenfunctions that are Dirac delta distributions, and thus not in the Hilbert space.) For example, if our wavefunctions are defined on a one-dimensional position space (ψ(x)), then the observable corresponding to the position is just multiplication by the position variable x: . One last important point to take away is the expectation value of an observable. In classical probability, the expected value of some function g of a random variable X is given by for a discrete random variable with probabilities Pi, and with probability density function P(x). Similarly, in quantum mechanics, we define: where * represents the complex conjugate. Note that for position operator , we see Which, given that |ψ(x)|2 gives our probability density, matches our standard definition. An example of these concepts in action can be seen in the proof of the Heisenberg uncertanty principle in this past post. […]

December 19, 2008 at 1:15 am |

[…] arises because two operators commute if and only if their commutator is zero. As demonstrated in this post, if two operators have a non-zero commutator, then they obey an uncertanty relation: . In fact, it […]