Chapter 8 Vector Spaces in Quantum Mechanics

Chapter 8 Vector Spaces in Quantum Mechanics 89a probability P(q n |ψ) of finding the system in the state |q n 〉 given it was in the state |ψ〉.Likewise, there will be a probability P(φ|q n ) of finding the system in the final state |φ〉given that it was in the state |q n 〉. It is at this point that we reach the result that, in a sense,defines quantum mechanics. If the possibility exists of finding the system in any of thestates |q 1 〉, |q 2 〉, . . . , then according to classical ideas of probability, we would be safe inassuming the intuitive formula∑P(φ|ψ) = P(φ|q n )P(q n |ψ) — a result of classical probability theoryni.e. the system starts in the state |ψ〉, and has the probability P(q n |ψ) that it will end upin the state |q n 〉, and once it ‘arrives there’ we then have the probability P(φ|q n ) that itwill then end up in the state |φ〉 given that it was in the state |q n 〉. Multiplying thesetwo probabilities will then give the probability of the system starting from |ψ〉 and passingthrough |q n 〉 on its way to the final state |φ〉. We then sum over all the possible intermediatestates |q n 〉 to give the total probability of arriving in the state |φ〉. However, what is foundexperimentally is that if the system is never observed in any of the intermediate states |q n 〉,this probability is not given by this classical result – the measurements show evidence ofinterference effects, which can only be understood if this probability is given as the squareof a ‘probability amplitude’.Thus, we find we must write P(φ|ψ) = |〈φ|ψ〉| 2 , where 〈φ|ψ〉 is a complex number, generallyreferred to as the probability amplitude of finding the system in state |φ〉 given that itwas in state |ψ〉, and this probability amplitude is then given by∑〈φ|ψ〉 = 〈φ|q n 〉〈q n |ψ〉 (8.51)nwhere the sum is over the probability amplitudes of the system passing through all thepossible states associated with all the possible observed values of the quantity Q.If we square this result we find that∑P(φ|ψ) = |〈φ|ψ〉| 2 = |〈φ|q n 〉| 2 |〈q n |ψ〉| 2 + cross termsn∑= P(φ|q n )P(q n |ψ) + cross terms. (8.52)nWe note that this expression consists, in part, of the classical result Eqn. (8.3), but there is,in addition, cross terms. It is these terms that are responsible for the interference effectsobserved in the two slit experiment, or in the Stern-Gerlach experiment. If we observewhich state |q n 〉 the system ‘passes through’ it is always found that these interferenceterms are washed out, reducing the result to the classical result Eqn. (8.3). This we haveseen in the case of two slit interference wherein if the slit through which the particlepasses is observed, then the interference pattern on the observation screen is washed out.What remains to be found is ways to calculate these ‘probability amplitudes’. We can goa long way towards doing this by recognizing, from a purely physical perspective, that theprobability amplitudes 〈q m |q n 〉 must have the following properties:• 〈q m |q n 〉 = 0 if m n. This amounts to stating that if the system is in the state|q n 〉, i.e. wherein the observable Q is known to have the value q n , then there is zeropossibility of finding it in the state |q m 〉.