PHYSICS 420 SPRING 2006 Dennis Papadopoulos. LECTURE 18. EXPECTATION VALUES QUANTUM OPERATORS. What have we learned from the Schrodinger Equation?. We can find allowed wave-functions.
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Alternatively, you can count the number of times you rolled a particular number and weight each number by the the number of times it was rolled, divided by the total number of rolls of the dice:
If you roll a dice 600 times, you can average the results as follows:
After a large number of rolls, these ratios converge on the probability for rolling a particular value, and the average value can then be written:
This works any time you have discreet values.
What do you do if you have a continuous variable, such as the probability density for you particle?
It becomes an integral….
The expectation value can be interpreted as the average value of x that we would expect to obtain from a large number of measurements. Alternatively it could be viewed as the average value of position for a large number of particles which are described by the same wave-function.
We have calculated the expectation value for the position x, but this can be extended to any function of positions, f(x).
For example, if the potential is a function of x, then:
If all xi the same s=0 and observable is sharp. Otherwise is fuzzy subject to the UP.
Found how to predict <x> and its position uncertainty Dx. Same for <U>.
How about p or KE?
We could do it if p was a function of position, i.e. p=p(x) was known. however in QM we cannot measure simultaneously x and p. Of course we can do it in classical physics since all observables are sharp and the uncertainty is related to measurement errors. In QM there is no path that connects p and x.
Need different approach. Identify <p> with <p>=m (d<x>/dt. Cannot be derived but guessed since it reduces to the correct classical limit.
expression for kinetic energy
kinetic plus potential energy gives the total energy
In general to calculate the expectation value of some observable quantity:
We’ve learned how to calculate the observable of a value that is simply a function of x:
But in general, the operator “operates on” the wave-function and the exact order of the expression becomes important: