Review of Quantum Mechanics, cont.

Transcription

1 Review of Quantum Mechanics, cont. 1 Probabilities In analogy to the development of a wave intensity from a wave amplitude, the probability of a particle with a wave packet amplitude, ψ, between x and x + dx is; P(x)dx = ψ (x)ψ(x) dx In general, ψ is complex. Then in momentum space the probability that a particle has momentum between k and k + dk is; p(k)dk = ψ (k)ψ(k) dk The configuration space function is related to the momentum function by a Fourier transformation; ψ(k) = 1 2π dxψ(x) e ikx Because the Schrodinger equation is linear (indeed the QM theory is constructed so that amplitudes add), then if ψ 1 and ψ 2 are solutions to the Schrodinger equation, aψ 1 + bψ 2 is a solution with a and b arbitrary, complex constants. Define the average value of a function with probability distribution P(x) as; f(x) = dxp(x) f(x) = dxψ (x)ψ(x) f(x) This is the expectation value of f(x) and can be easily generalized to more than one dimension. The probability function must satisfy the several characteristics of all probability distributions. It must be normalizable, i.e. dxp(x) = (finite number) which is set to unity. It must be such that P(x) 0. Finally, it must be conserved. To show the later, consider the following operation; P(x) = [ ψ (x) ψ(x) + ψ (x) ψ(x) ] Apply the Schrodinger equation, i ψ P(x) = 2 /2m 2 ψ x 2 to obtain; = i /2m x (x) [ ψ ψ(x) ψ x (x) ψ(x) x ] Identify a probability current; 1

2 S(x) = i /2m x (x) [ ψ x and write; P + S x = 0. This takes the general form in 3-D; P + S = 0; ψ(x) ψ (x) ψ(x) x ] which is the equation of continuity for the probability. Recall that S represents the flux of vector, S, out of a volume per unit volume. Suppose we integrate this equation over all space. In 1-D, for example, this becomes; dxp(x) = (i /2m) ψ x (ψ x ψ ψ x ) dx = S n The probability current perpendicular to the boundary surface, S n = S n, vanishes, so probability is conserved if the system is isolated. 2 Correspondance principle One of the most powerful ways to investigate a problem is to search for solutions in limiting cases, eg allowing a variable to approach a limit, for example or 0. In many such cases the complexity of a problem is reduced and the physics becomes more obvious. This provides a technique to examine whether a solution is correct or even whether the physics applies to the problem at hand. The QM correspondance principle states that the laws of QM must be chosen such that they approach their classical counterpart in some limit. In particular, take the limit as h 0 which should result in the classical result. 3 Deterministic laws, hidden variables, and QM Classical physics relies on statistical calculations in many situations. A good case in point is thermodynamics, where variables such as temperature or pressure are averages over the random motion of the individual particles. The large number of substructures means that statistical fluctuations in such a measurement are very small. However, the point here is that classical physics assumes that a system is deterministic in the sense that if the laws of mechanics were applied to all of the substructures, then the entire dynamics of the composite 2

3 D+ a b D+ S A D D B CM Figure 1: The layout of a simple two-channel test of Bell s theorem system could be determined. Statistics in this sense is a convenience which avoids some very difficult or involved calculations. This concept, if carried over into QM, is not correct as QM does not have deterministic underpinnings in the sense that there are no hidden variables. This was first outlined by the Einstein, Podolsky and Rosen paradox which Einstein said implied spooky action-at-adistance. That is, information which travel faster than c. QM predictions are based on probability, but classical probability implies that a particle, for example, can have a definite position in space and time. However, QM lacks the ability to predict such a value. If this were true, then QM would be incomplete as there would be hidden variables which have been removed by averaging. Einstein (and others) worked for 20 years ( ) in an attempt to understand this paradox. The crux of the problem lies in the assumption of local realism and is reflected in what today is called quantum entanglement. A local realistic theory assumes; 1. Objects have a definite localized state which determines all measurable parameters 2. Information of local actions cannot travel faster than c. Thus if two systems are sufficiently far apart, they can have no effect on each other Either QM or local realism is wrong. Classical probability and the predictions of QM are incompatable, as demonstrated by the Bell inequality theorem. which is amendable to experimental verification. A schematic two channel Bell test is shown in Figure 1. In the figure, the source, S, emitts pairs of photons moving to detectors A and B. Each photon passes through a polarizer which may be set by experimenters A and B. The emerging signals are detected and counted in coincidence in CM. The experiment counts the polarization 3

4 values + or - in coincidence yielding the results of + +, + -, - +, or - -. The polarimeters, a, and b are set at angles 0, 45, 22.5, Now if local hiddden variables are acting, they encode the photon polarizations at the time of their emission. If A and B measure the polarization along the same axis the correlation is 100%. If A and B measure at right angles the correlation is 0. Suppose A and B can rotate their apparatus relative to each other by any amount at any time, even after the photons are emitted. Upon averaging over a number of trials, hidden variables would produce a polarization result of 50%. However, in QM the photons are entangled and the predicted correlation depends on the correlation function, cos(θ). Working through the projections of the correlations one finds a result of It appears that when A makes a measurement B somehow instantaneously finds a result which increases the correlation. According to QM the photons are entangled ( local realism limits this correlation), and the particle properties are only defined after a measurement is made on either particle. This has led to the field of quantum information theory, and is the basis of quantum crypotography. 4 Example of a bound state Wave packets in a bound system spread throught the system space. A non-relativistic wave packet can be found by solving the Schrodinger equation. This equation has the form, E = T + V where E and T are energy and momentum operators. The potential energy, V, is a function of the corrdinates. An alternative equation in momentum space can be obtained by Fourier transform. Here I want to introduce a bound state by choosing a potential form; V (x) = V 0 (a 2 r 2 ). This potential form (harmonic oscillator) is shown in Figure 2. It is unrealistic for large r, but because an analytical solution can be obtained, and any potential form has similar behavior for small r, the solution will illustrate several important concepts. The potential form is widely used. The Schrodinger equation is then; E = [ 2 2m 2 + V 0 r 2 ]ψ Here the energy operator i ψ has been evaluated as the energy, E + V 0a 2, and the equation is written in 3-D. To begin, we solve the 1-D equation 2 2 x 2. The equation is an eigenvalue problem, so that it has solutions only for a quantized set of functions and energies. Make the substitution χ = αx to obtain; 4

5 V(r) a r V a 2 o V = V o ( a 2 r 2 ) Figure 2: The harmonic oscillator potential d 2 ψ dχ 2 + (λ 2 χ 2 )ψ = 0 where α 4 = 2mV 0 2 and λ 2 = 2E ( α) 2. Then assume a solution of the form; ψ = e χ/2 n a n χ n+s Substitute and collect terms in powers of χ a n (n + s)(n + s 1)χ n+s 2 2 n n χ n+s + λ 2 a n χ n+s = 0 n n a n (n + s)χ n+s Because each power of χ is linearly independent the coefficient can be set equal to zero. For example for n = 0 and for n = 1 s(s 1)a 0 = 0 s(s + 1)a 1 = 0 The general term is found to be 5

6 (s + n + 2)(s n + 1)a n+2 (2s + 2n + 1 λ 2 )a n = 0 However this series does not converge if n so that it must terminate by requiring λ 2 = 2s + 2n + 1. The requirement that the solution remain finite provides an infinite set of solutions (eigenfunctions) with corresponding values of discrete energies (eigenvalues λ 2 = ω 2E with ω c = ( 2V 0 c m )1/2. Now choose; s = 1 or 0 Then the energy is quantized with eigenvalues; resulting in; λ 2 = 2n + 1 or 2n + 3 E = (n + 1/2) ω c The eigenfunctions for several values of n are shown in Figure 3. The eigenvalues of high n are separated by an small energy interval when compared to the total energy, so that classical motion is approached. Thus we find that the system is more probably found at its turning points where the velocity of the wave amplitude is near zero. The fraction of the time that a classical oscillator spends in a given time interval is found to be; dt T = (1/T) 2dx ω c x 2 0 x 2 where T is the period and x 0 is the turning point of the oscillation. Note also that the solution decreases as the negative exponential of x 2, and the zero point energy occurs when n = 0. The normalized wave function for n = 0 has the form; ψ 0 = (1/ π) e x2 /2 The wave function cannot be confined to a point, but has a finite width, x. From the uncertainty principle there is an uncertainty in the momentum x p = and apply Eψ = [ 2 2m 2 + V 0 x 2 ]ψ. to obtain; E = p2 2m + mω2 c 2 2 p 2 Minimize this energy by E p and find that E = ω c /2 6

7 Figure 3: Harmonic oscillator eigenfunctions for several values of N 7

8 Now there are 2 distinct types of solutions characterized by the selection of the starting integer s. These solutions are either odd or even in reflections about the origin, x = 0. This degeneracy is called parity. Note that the potential energy function depends on the square of the position, x 2, which makes the energy eignevalue independent of its parity. In 1-D, parity is the symmetry operation, x x. The harmonic oscillator is invarient under the symmetry of parity. 8