LECTURE 18

EXPECTATION VALUES

QUANTUM OPERATORS

PHYSICS 420SPRING 2006Dennis Papadopoulos

•We can find allowed wave-functions.

•We can find allowed energy levels by plugging those wavefunctions into the Schrodinger equation and solving for the energy.

•We know that the particle’s position cannot be determined precisely, but that the probability of a particle being found at a particular point can be calculated from the wave-function.

•Okay, we can’t calculate the position (or other position dependent variables) precisely but given a large number of events, can we predict what the average value will be? (If you roll a dice once, you can only guess that the number rolled will be between 1 and 6, but if you roll a dice many times, you can say with certainty that the fraction of times you rolled a three will converge on 1 in 6…)

SHARP AND FUZZY OBSERVABLES

• Two types of measurable quantities associated with or– Sharp: e.g. Energy for stationary states. Every

measurement performed gives the same value controlled by the quantum labeling of the wave n.

– Fuzzy: e.g. position or momentum. A partiple described by can have occupy different places and have different momentum, with a probability given by *. Predictions for can be tested by making repeated measurement of the quantity.

If you roll a dice 600 times, you can average the results as follows:

600

...43556421 ...)3(

600

104)2(

600

97)1(

600

99

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:

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:

xxPx

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….

2

( , )xx xP dx x x t dx

Table 6-1, p. 217

2

2.5 3.7 1.4 .... 5.35.46

181.4(1/18) 2.5(1/18) ... 5.4(3 /18) 6.2(2 /18) ... 8.8(1/18) 5.46

,

( , )

x

x

x xP

Expectation value

x x x t dx

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:

dxtxxUU

2),()(

dxtxxUU

2),()(

UNCERTAINTY AND STADARD DEVIATION

2

2

22 2

( , )

( ) ( , )

( , )

x x x t dx

f f x x t dx

x x x t dx

Standard Deviation

2

1

2

2 22 21

1 1

22

22

22

( )

( )2 ( ( ) / ) (1/ ) 2

N

i

N

i N N

i

x x

N

xx x N x N x x x x

N

x x

x x

x x x

If all xi the same =0 and observable is sharp. Otherwise is fuzzy subject to the UP.

QUANTUM OPERATORS

Found how to predict <x> and its position uncertainty x. 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.

*

*

{ [ ( , ) ( , )] }

( , )( , )( )

d x dp m m x x t x t dx

dt dtSE

x tp x t dx

i x

h

Momentum operator

i x

h

the potentialexpression for kinetic energy

kinetic plus potential energy gives the total energy

2

;2

pKE p k

m h

2

;2

pKE p k

m h

position x x

momentum p

potential energy U U(x)

kinetic energy K

total energy E

i x

h

2 2

22m x

h

it

h

observable

operator

dxQQ

*

dxxUdxxUdxUU2

** )()(

2 2

* *22

K K dx dxm x

h

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:

Table 6-2, p. 222