By Shen S.-Q.
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Example text
De Broglie waves play an appreciable role, therefore, only in the realm of subatomic particles. The response of the wave properties of a particle to an external force follows a basic law of quantum mechanics that, in its mathematical form, is known as the Schrödinger equation. 9 The Uncertainty Relation The commutation relation [x> p] = l~ shows that the position and the momentum are incompatible. This relation leads to the famous uncertainty relation (x)2 ® ® (p)2 ~2 @4 ® Let A ad B be observables.
3 Time Dependence of Expectation Value: Spin Precession. It is instructive to study how the expectation value of an observable changes as a function of time. To end this we treat an example here: spin precession. , |±i are also the energy eigenkets with eigenvalues ±~$@2= Suppose at w = 0 |i = F+ |+i + F |i 52 CHAPTER 2 – MANUSCRIPT At a later time t, |> w0 : wi = X (w> w0 ) |i = F+ h l$w 2 |+i + F h+ l$w 2 |i Specifically, (1) Let us suppose the initial state |i is the spin up state |+i >i,e.
I = P ¯¯ (q) ® (q) ® d d | ). The expectation value of A with respect to state |i is defined as hAi h |A| i = = P q (q) d E ED P D (q) E D (q) |d d |A| d(p) (p) |d q=p ¯D E¯2 P ¯ (q) ¯ × ¯ |d ¯ = d(q) q q ¯ ® where q is the probability in the state ¯d(q) . Selective measurement (or filtration) 20 CHAPTER 1 – MANUSCRIPT We imagine a measurement process with a device that selects only one of the eigenket and rejects all others. Mathematically, we can say that such a selective measurement amounts to applying the projection operator.
Lecture notes on quantum mechanics by Shen S.-Q.
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