A Crash Course on Quantum Mechanics
A Crash Course on Quantum Mechanics A Crash Course on Quantum Mechanics
The Copenhagen interpretation follows the second one. As a result, before actuallymeasuring A, you have no way of predicting which outcome will be the obtained. Also,any one of them with nonzero d n can really occur. During the experiment, nature somehowdecides which one should appear. This lack of determinism terrified a lot of people.Einstein was their leader. Bohr was the defender of the interpretation. After a lot ofdiscussion, this view gained weight. But even today there are serious works concentratingon other possible interpretations.Collapse. To complete the theory, we have to mention one last feature. Consider anexperiment where you measure A. After some indeterministic measurement process we getone particular result, λ n . Now, suppose that we measure A again immediately after thefirst measurement. We need to do this immediately because with time, the wavefunctioncould change. Normally we should obtain exactly the same result. In other words, bothexperiments should give the same value λ n . For a correct measurement concept we needto have this feature. For this reason, the second experiment has no uncertainty in it. Ifthis is so, then the first measurement of A should have caused a discontinuous change inthe wavefunction to the eigenfunction of  corresponding to λ n.To summarize, suppose that you make a measurement at time t = t meas . Prior themeasurement, the wavefunction isψ(⃗r, t = t meas − ɛ) = ∑ md m α m (⃗r) ,which can be anything. If the measurement yields A = λ n , then the wavefunction justafter the measurement has to beψ(⃗r, t = t meas + ɛ) = α n (⃗r) .In other words, the effect of the measurement on the wavefunction is a projection to aneigenfunction (or eigenspace) of  and re-normalization. This is called the collapse of thewavefunction.It can be seen that the measurement introduces an unavoidable change in the wavefunction.A change that destroys all information that is carried by the wavefunction beforethe measurement. So, measurement in quantum mechanics does not have the conventionalmeaning of “learning”, its meaning would be more like “changing and pretending that youhave learned”.State. In quantum mechanics, the wavefunction ψ(⃗r, t) contains all information that youcan ever learn about the particle. For this reason, it is frequently referred as the state. Forexample, if you know the precise state at a certain time, then you can calculate the state atany other time by integrating the Schrödinger equation. The corresponding notion of statein classical mechanics would be its position and momentum: (⃗r, ⃗p), which is a point in thesix dimensional phase space. In quantum mechanics, however, the state space becomes aninfinite dimensional Hilbert space.Consider now the position property of particle in a state ψ(⃗r). Since the wavefunctionis distributed in space, there is no single definite position which we can say the particle is16
located. Measurement of position gives us only one of these possibilities, but before themeasurement each position is a possibility.There is an interpretation which you might hear a lot which goes like this: “Particleis actually somewhere but we don’t know where it is”. This sentence actually assumesthat state=(ψ, ⃗r real ), where ⃗r real is the supposed real position of particle. In other words,it assumes that there are more things to know about the particle than the wavefunction.This is a hidden variable theory and is entirely different from quantum mechanics.In quantum mechanics we might state the same thing as “particle is everywhere” which,at first sight, might look confusing. Another alternative statement is “there is no meaningof question ‘where is it?’ without actually measuring it”. It appears that the classicalnotions of definite position and definite momentum cannot be directly carried over toquantum mechanics. We have notions of position and momentum in quantum mechanicsbut their nature is different from our classical notions.This situation is similar to the notion of absolute time we meet in relativity. Absolutetime is a concept which appears to be true in the non-relativistic limit. Of course such anotion is invalid, and we will make a lot of mistakes if we try to directly carry it over torelativistic problems. Same in quantum mechanics. We should, then, get rid of the notionsof definite values of some mechanical quantities.Uncertainties. We don’t have definite values of position and momentum, but there isalso a limitation on how close we can get to definiteness. A mathematical measure of thisis the standard deviation of measurement results which is frequently called uncertainty.For example, the uncertainty in x-component of position is∆x 2 = 〈 (x − 〈x〉) 2〉 ∫= |x − 〈x〉| 2 |ψ| 2 .Remember that this gives the deviation of measurement results from the average in a seriesof repeated measurements on the same state ψ. In other words, each time the particle hasto be re-prepared in the same state. Uncertainty in momentum is defined similarly,∆p 2 x = 〈 (p x − 〈p x 〉) 2〉 .Now, it appears that the position and momentum operators corresponding to the samecomponent do not commute with each other. They have a commutatorˆxˆp x − ˆp xˆx = i¯h .This commutation relation implies that the product of respective uncertainties has a lowerbound. The proof goes like this. First note that∆x 2 = 〈ψ| (ˆx − 〈x〉) 2 ψ〉= 〈(ˆx − 〈x〉) ψ| (ˆx − 〈x〉) ψ〉 .Let us define two vectors in the Hilbert space byφ 1 = (ˆx − 〈x〉)ψφ 2 = (ˆp x − 〈p x 〉)ψ17
- Page 7: This relation is called Bohr freque
- Page 11: which is nothing other than the tim
- Page 15: which implies that ˆp x = ˆp †
- Page 19 and 20: Postulate 1: States. For every isol
located. Measurement of positi<strong>on</strong> gives us <strong>on</strong>ly <strong>on</strong>e of these possibilities, but before themeasurement each positi<strong>on</strong> is a possibility.There is an interpretati<strong>on</strong> which you might hear a lot which goes like this: “Particleis actually somewhere but we d<strong>on</strong>’t know where it is”. This sentence actually assumesthat state=(ψ, ⃗r real ), where ⃗r real is the supposed real positi<strong>on</strong> of particle. In other words,it assumes that there are more things to know about the particle than the wavefuncti<strong>on</strong>.This is a hidden variable theory and is entirely different from quantum mechanics.In quantum mechanics we might state the same thing as “particle is everywhere” which,at first sight, might look c<strong>on</strong>fusing. Another alternative statement is “there is no meaningof questi<strong>on</strong> ‘where is it?’ without actually measuring it”. It appears that the classicalnoti<strong>on</strong>s of definite positi<strong>on</strong> and definite momentum cannot be directly carried over toquantum mechanics. We have noti<strong>on</strong>s of positi<strong>on</strong> and momentum in quantum mechanicsbut their nature is different from our classical noti<strong>on</strong>s.This situati<strong>on</strong> is similar to the noti<strong>on</strong> of absolute time we meet in relativity. Absolutetime is a c<strong>on</strong>cept which appears to be true in the n<strong>on</strong>-relativistic limit. Of course such anoti<strong>on</strong> is invalid, and we will make a lot of mistakes if we try to directly carry it over torelativistic problems. Same in quantum mechanics. We should, then, get rid of the noti<strong>on</strong>sof definite values of some mechanical quantities.Uncertainties. We d<strong>on</strong>’t have definite values of positi<strong>on</strong> and momentum, but there isalso a limitati<strong>on</strong> <strong>on</strong> how close we can get to definiteness. A mathematical measure of thisis the standard deviati<strong>on</strong> of measurement results which is frequently called uncertainty.For example, the uncertainty in x-comp<strong>on</strong>ent of positi<strong>on</strong> is∆x 2 = 〈 (x − 〈x〉) 2〉 ∫= |x − 〈x〉| 2 |ψ| 2 .Remember that this gives the deviati<strong>on</strong> of measurement results from the average in a seriesof repeated measurements <strong>on</strong> the same state ψ. In other words, each time the particle hasto be re-prepared in the same state. Uncertainty in momentum is defined similarly,∆p 2 x = 〈 (p x − 〈p x 〉) 2〉 .Now, it appears that the positi<strong>on</strong> and momentum operators corresp<strong>on</strong>ding to the samecomp<strong>on</strong>ent do not commute with each other. They have a commutatorˆxˆp x − ˆp xˆx = i¯h .This commutati<strong>on</strong> relati<strong>on</strong> implies that the product of respective uncertainties has a lowerbound. The proof goes like this. First note that∆x 2 = 〈ψ| (ˆx − 〈x〉) 2 ψ〉= 〈(ˆx − 〈x〉) ψ| (ˆx − 〈x〉) ψ〉 .Let us define two vectors in the Hilbert space byφ 1 = (ˆx − 〈x〉)ψφ 2 = (ˆp x − 〈p x 〉)ψ17
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