Operators•A function is something that turns numbers into numbers•An operator is something that turns functions into functions

•Example: The derivative operator

sinf x kxd

dxO =

( ) ( )df x f x

dxO =

•In quantum mechanics, x cannot be the position of a particle•Particles don’t have a definite position

•Instead, think of x as something you multiply a wave function by to get a new wave function

•x is an operator, sometimes written as xop or X•There are lots of other operators as well, like momentum

( ) ( )opx x x xy y=

p ki x

oppi x

( ) ( )sin cosd

kx k kxdx

= =

Expectation Values•Suppose we know the wave function (x) and we measure x. What answer will we get?

•We only know probability of getting different values•Let’s find the average value you get

•Recall |(x)|2 tells you the probability density that it is at x•We want an expectation value

•It is denoted by x

x

x P x x 2

x x xdx

* x x x dx

•For any operator, we can similarly get an average measurement

* x x dxO O

op*p x p x dx

*x

x dxi x

Sample ProblemA particle is in the ground state of a harmonic oscillator.

What is the expectation value of the operators x, x2, and p?

2140 2exp

Ax Ax

*x x dx

122

12 if is even

0 if is odd

nn

n Ax A nx e dx

n

2 2*x x dx

2 1 2x A

0x

*p dxi x

0p

Note: x2 x2More on this later

Note: Always use normalized wave functions for expectation values!

mA

31 12 2 2,

2 22 2Ax AxAe Ax e dx

i

2 2AxA e x dx

2AxA e xdx

The Hamiltonian Operator•In classical mechanics, the Hamiltonian is the formula for energy in terms of the position x and momentum p•In quantum, the formula is the same, but x and p are reinterpreted as operators

2

2

pH V x

m

2op

op

2 2

op2

2

2

pH V x

m

V xm x

•Schrodinger’s equations rewritten with the Hamiltonian: E H

i Ht

Advanced Physics:•The Hamiltonian becomes much more complicated

•More dimensions, Multiple particles, Special Relativity•But Schrodinger’s Equations in terms of H remain the same

•The expectation value of the Hamiltonian is the average value you would get if you measure the energy

E H

Sample ProblemA particle is trapped in a 1D infinite

square well 0 < x < L with wave function given at right. If we measure the energy, what is the average value we would get?

25

30x Lx x

L

E H 2

*2

oppV x dx

m

2 2

20

*2

L ddx

m dx

2 2

2 25 2

0

30

2

L dLx x Lx x dx

mL dx

2

25

0

152

L

Lx x dxmL

2

2 31 12 35 0

30 L

Lx xmL

23 31 1

2 35

30L L

mL

2

2

5

mL

•Compare to ground state:•Often gives excellent approximations

2 2 2

2 2

4.935

2E

mL mL

Tricks for Finding Expectation Values•We often want expectation values of x or x2 or p or p2

•If our wave function is real, p is trivial

*d

p dxi dx

d

dxi dx

2

2

ddx

i dx

2

2i

0p •To find p2, we will use integration by parts2 2

22

*d

p dxi dx

2 **

d d ddx

dx dx dx

22 2 d

p dxdx

•Recall: x2 x2. Why?•The difference between these is a measure of how spread out the wave function is•Define the uncertainty in x:

Uncertainty

22 24 6 4 626 25

2 2

22 2x x x

•We can similarly define the uncertainty in any operator:

22 2p p p 22 2O O O

Heisenberg Uncertainty Principle

12x p

Sample ProblemA particle is in the ground state of a harmonic

oscillator. Find the uncertainty in x and p, and check that it obeys uncertainty principle

2 1 2x A0x 0p •Much of the work was done five slides ago•We even found p, but since is real, it is trivial anyway•Now work out p2:

2 212p A

•Now get the uncertainties

22x x x 1

2A

22p p p 2A

12x p

2140 2exp

Ax Ax

mA

22 2 d

p dxdx

2

2 212exp

AAx Ax dx

22 2 2 AxA A x e dx