Wavefunctions and Expectation Values

Schrodinger s picture of quantum mechanics describe any object (for example, an electron) by its wavefunction, fix). The wavefunction itself is not directly observable, but it contains information about all possible observations because of the following two properties. 

i/r (x)i/r(x) = 11/0 (x) 2 = / (x), the probability of finding the object at position x. P(x) 0 by definition, but the wavefunction is not just the square root of the probability. Wavefunctions at any point can be positive, negative, or even complex. This phase variation of the wavefunction is central to quantum mechanics. It lets particles exhibit wave-like behavior such as interference at positions where two waves are out of phase, just as classical waves exhibit interference (Chapter 3). 

Since the object must be somewhere, with probability 1, 

All possible wavefunctions are continuous (no breaks or jumps) and satisfy Equation 6.4. 

Given any observable quantity A (for example, the position or momentum of the object), the wavefunction (/ (.r ) lets us calculate the expectation value (A) which is the average value you would get if you made a very large number of observations of that quantity. The wavefunction thus contains all the information which can be predicted about the system. 

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