關於量子力學的測不準定理

The precise statement of the uncertainty principle is summarized by the formula

圖片參考：http://imgcld.yimg.com/8/n/HA05726829/o/701007100065713873367410.jpg

where A and B are physical observables (position, momentum, energy, time, spin, etc), bracket denotes expectation value (mean value), dA is the deviation of A from its mean value, [A,B] = AB - BA is the commutator (see examples below).

First of all, the uncertainty principle says that A and B cannot take certain values simultaneously if the commutator is non-zero. Otherwise, this would violate the above equation by setting the right side to zero. This has NOTHING TO DO WITH PHOTONS.

Example 1 : Momentum and position
The commutator of momentum and position is [p,x] = - i hbar, where hbar = h / 2 pi, and h is the Planck's constant.

圖片參考：http://imgcld.yimg.com/8/n/HA05726829/o/701007100065713873367411.jpg

This says p and x cannot be measured simultaneously.

Example 2 : Spin
Let S_x, S_y and S_z be the spin (angular momentum) operators along x, y and z direction. For spin-1/2 particles such as electrons, the commutator is [S_x,S_y] = hbar S_z.

圖片參考：http://imgcld.yimg.com/8/n/HA05726829/o/701007100065713873367422.jpg

This says generically spin along different directions cannot be measured simultaneously.

Back to your question. Any measurement or observation will in general change the physical state. Before measurement, a quantum state can have uncertainty in quantities A and B. After a measurement in one of the variables, say A, the quantum state has certain fixed value in A. And B will take the highest uncertainty (infinity).

For example, a state of a moving particle is describe quantum mechanically by a wave packet that has both uncertainty in position and momentum. Now let's measure its momentum. Say it is observed with momentum p. Then the state is now determined with certain momentum p. The wave packet is changed into a plane wave exp(i p x / hbar).

For spins, a state is in general a spin state (up or down) along some unknown direction. Let's measure it along the z-direction. The observation can either take +hbar/2 or -hbar/2 corresponding to spin up or down along z-axis. An observation therefore align the state in the z-direction.

Take an electron for example. If we need to determine the position and momentum of an electron, we have to use a photon to detect it. But when a photon interacts with an electron, Compton scattering takes place. The electron absorbs part of the energy of the photon, and scattered away by the photon, which is now at a lower energy. The momentum of the electron thus has altered as well as its position.

As we all know that Compton scattering depends on the energy of the photon and the "angle of impact" (similar to the oblique collision of two billard balls in classical mechanics) at that instant. The energy transfer to the electron thus cannot be pre-determined. In other words, this is a process of probability.