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P on the screen from the slit , and we multiply these two to get the amplitude a s, t ; xa t, for it to reach the screen at p via t. as with probabilities, this is the correct amplitude, assuming the top slit is open, whether or not the bottom slit b is open. similarly, assuming that b is open, there is an amplitude a s, b ; xa b, for the photon to reach p from s via b whether or not t is open ; . if both slits are open, then we have a total amplitude a x a b, for the photon to reach p from s. this is all very well, but it is not of much use to us until we know how to interpret these amplitudes when a quantum effect is magnified until it reaches the classical level. we might, for example, have a photon detector, or photo-cell placed at p, which provides a way of magnifying an event at the quantum level the arrival of a photon at p into a classically discernible occurrence, say an audible 'click'. if the screen acts as a photographic plate, so that the photon leaves a visible spot, then this would be just as good, but for clarity let's stick to using a photo-cell. ; there must be an actual probability for the 'click' to happen, not just one of these mysterious 'amplitudes'! how are we to pass from amplitudes to probabilities when we go over from the quantum to the classical level? it turns out that there is a very beautiful but mysterious rule for this. the rule is that we must take the squared modulus of the quantum complex amplitude to get the classical probability. what is a 'squared modulus'? recall our description of complex numbers in the argand plane chapter 3, p. 117 ; . the modulus \z\ of a complex number z is simply the distance from the origin i. e. from the point 0 ; of the point described by z. the squared modulus |z|2 is simply the square of this number. thus, if z x + iy, where x and y are real numbers, then by the pythagorean theorem, since the line from 0 to z the hypotenuse of the right-angled triangle 0, x, z ; our required squared modulus is [zl + y2. 309 fig. 6. 8. a probability amplitude represented as a point z within the unit circle in the argand plane. the squared distance |z[2 from the centre can become an actual probability when effects become magnified to the classical level. note that for this to be an actual 'normalized' probability, the value of |z|2 must lie somewhere between 0 and 1. this means that for a properly normalized amplitude, the point z in the argand plane must then lie somewhere in the unit circle see fig. 6. 8 ; . sometimes, however, we shall want to consider combinations w x alternative a + z alternative b, where w and z are merely proportional to the probability amplitudes, and then they need not lie in this circle. the condition that they be normalized and therefore provide actual probability amplitudes ; is that their squared moduli should sum to unity: h2 + |z|2 i. if they are not so normalized, then the actual amplitudes for a and b, respectively, are m v |w|2 + |z|2 ; and z v |w|2 + |z|2 ; , which do lie in the unit circle. we now see that a probability amplitude is not really like a probability after all, but rather more like a 'complex square root' of a probability. how does this affect things when effects at the quantum level get magnified so as to reach the classical level? recall that in manipulating probabilities and amplitudes we sometimes needed to multiply them and we sometimes needed to add them. the first point to note is that the operation of multiplication poses no problem in our passing from the quantum to the classical rules. this is because of the remarkable mathematical fact that the squared modulus of the product of two complex numbers is equal to the product of their individual squared moduli: \zw h2 ! 2. this property follows at once from the geometrical description of the product of a pair of complex numbers, as given in chapter 3; but in terms of real and imaginary parts z x + iy, w u iv, it is a pretty little miracle. try it! ; this fact has the implication that if there is only a single route open to a particle, e. g. when there is only a single slit say ; open, in the two-slit experiment, then one can argue 'classically' and the probabilities come out the same whether or not an additional detection of the particle is made at the intermediate point at ; . * we can take the squared moduli at both stages or only at the end, e. g. |a s, ; |2 , and the answer comes out the same each way for the resulting probability. however, if there is more than one route available e. g. if both slits are open ; , then we need to form a sum, and it is here that the characteristic features of quantum mechanics begin to emerge. when we form the squared modulus of the. 71 ; international business machines corporation [us us]; new orchard road, armonk, ny 10504 us ; . 71 ; ibm united kingdom limited [gb gb]; p.o. box 41, north harbour, portsmouth, hampshire po6 3au gb ; . only for seulement pour mg ; 72 ; chu, jack, oon; 44 shelbourne lane, manhasset hills, ny 11040 us ; . dimilia, david; 14 marion avenue, wappingers falls, ny 12590 us ; . huang, lijuan; 98 hillside place #2, tarrytown, ny 10591 us ; . 74 ; ling, christopher, john; ibm united kingdom limited, intellectual property law, hursley park, winchester, hampshire so21 2jn gb ; . 81 ; zw. 84 ; ap gh published publie : c ; 51 ; h01l 21 768, 21 ; wo 02 33747 21 ; pct jp01 08623 22 ; 1 oct oct 2001 01.10.2001 ; 25 ; ja 30 ; 2000-317661 26 ; ja 18 oct oct 2000 18.10.2000 ; jp 13 ; a1. Given dialysis without her knowledge and the patient nearly died, freilich said. she was particularly vocal about uninsured and disabled patients receiving less care than other patients. but her advocacy did not sit well with harford administrators. when freilich applied for the standard two-year renewal of her credentials in 1998, the hospital balked, agreeing only to a one-year renewal. in april 2000, the hospital board, even though it did not question her clinical competence, said it would not reappoint her. the denial of privileges "was based upon your failure to demonstrate ethical and cooperative behavior with regard to your position in the hospital and patient care, " according to a letter freilich later received from the maryland department of health and mental hygiene. the state launched its own investigation of freilich. four months later, she was cleared. but by then, the hospital had submitted a report on her to the national practitioner data bank, a list that is kept to identify doctors with malpractice judgments or who have lost their hospital privileges because of misconduct. freilich's listing flagged her as a problem physician to any future employer. she has not worked in her home county since and has struggled to keep her practice alive. after freilich filed suit in december 2000, the hospital's attorneys countered that if she were successful, "all disruptive, yet clinically competent, physicians would have been insulated from peer review in maryland hospitals." "what grabbed my attention, " said ullberg, 70, "was that she is a good practitioner and yet the system was beating up on her." althoug freilich is still pursuing a lawsuit in state court, ullberg believes the chances of any successful legal challenge to the law "are probably low." that's bad news for freilich and other physicians like her, ullberg said, but it's also bad news for patients. with reimbursements falling short of costs, hospitals will continue to face financial pressure to lower expenses, he believes, and eventually that will mean lowering the acceptable minimum standard of care. if a doctor notices this, he might not speak up under the present law, ullberg said, because if he is threatened with the loss of his credentials "he can't afford to care if a patient lives or dies.

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