4.3 measurement and expectation values - | stanford lagunita...nnn n cte r ... measurement and...

4.3 Measurement and expectation values

Slides: Video 4.3.1 Quantum-mechanical measurement

Text reference: Quantum Mechanics for Scientists and Engineers

Section 3.8

Measurement and expectation values

Quantum-mechanical measurement

Quantum mechanics for scientists and engineers David Miller

Probabilities and expansion coefficients

Suppose we take some (normalized) quantum mechanical wave function

and expand it in some complete orthonormal set of spatial functions

At least if we allow the expansion coefficients cn to vary in time

we know we can always do this

,t r

n r

, n nn

t c t r r

Probabilities and expansion coefficients

Then the fact that is normalized means we know the answer for the normalization integral

Because of the orthogonality of the basis functions only terms with survive the integration

Because of the orthonormality of the basis functions the result from any such term will simply be

Hence we have

,t r

2 3 3, 1n n m mn m

t d c t c t d

r r r r r

n m

2nc t

2 1nn

c

Measurement postulate

On measurement of a statethe system collapses into the nth

eigenstate of the quantity being measuredwith probability

In the expansion of the state in the eigenfunctions

of the quantity being measuredcn is the expansion coefficient

of the nth eigenfunction

2n nP c

Expectation value of the energy

Suppose do an experiment to measure the energy E of some quantum mechanical system

We could repeat the experiment many timesand get a statistical distribution of results

Given the probabilities Pn of getting a specific energy eigenstate, with energy En

we would get an average answer

where we call this average the “expectation value”

2n n n n

n n

E E P E c E

Energy expectation value example

For example, for the coherent state discussed above with parameter N, we have

where we use the result that the average in a Poisson statistical distribution is just the parameter N

Note that N does not have to be an integerThis is an expectation value, not an eigenvalue

We can have states with any expectation value we want

0 0

exp exp 1 1! ! 2 2

n n

nn n

N N N NE E n N

n n

Stern-Gerlach experiment

This apparatus has a non-uniform magnetic fieldlocally stronger near the

North pole magnet facebecause it is narrower

N

S

magnet North pole

magnet South pole

screen

magnetic field more

concentrated here

than here


Imagine firing some small magnets initially along the dashed line

Because the magnetic field is non-uniformstronger near the North

pole than near the South polea vertical magnet will be

deflected up

N

S

N

S

magnet North pole

magnet South pole

screen


Imagine firing some small magnets initially along the dashed line

Because the magnetic field is non-uniformstronger near the North

pole than near the South polea vertical magnet will be

deflected up or down

S

N

N

S

magnet North pole

magnet South pole

screen


A horizontally-oriented magnet will not be deflected

SN

N

S

magnet North pole

magnet South pole

screen


A horizontally-oriented magnet will not be deflectedand magnets of other

orientationsshould be deflected by intermediate amounts

After “firing” many randomly oriented magnetswe should end up with a line

on the screen

N

S

magnet North pole

magnet South pole

screen

?

Electrons and the Stern-Gerlach experiment

Electrons have a quantum mechanical property called spin

It gives them a “magnetic moment”just like a small magnet

What will happen if we fire electronswith no particular “orientation” of their spin

into the Stern-Gerlach apparatus?We might expect the “line” on the screen

(Note: the actual experiment used silver atoms, which behave the same as electrons in this case)


With electronswe get two dots!

“Explanation”We are measuring the vertical

component of the spinThere are two eigenstatesof this componentup and down

so we have collapse to the eigenstates

N

S

magnet North pole

magnet South pole

screen

electrons


Slides: Video 4.3.3 Expectation values and operators


Sections 3.9 – 3.10


Expectation values and operators


Hamiltonian operator

In classical mechanics, the Hamiltonian is a function of position and momentum

representing the total energy of the systemIn quantum mechanical systems that can be analyzed by

Schrödinger’s equationwe can define the entity

so we can write the Schrödinger equations as

and

2ˆ ,2

H V tm

2

r

,ˆ ,t

H t it

r

r H E r r


The entityis not a number

is not a functionIt is an “operator”

just like the entity is a spatial derivative operator

We will use the notation with a “hat” above the letter to indicate an operator

The most general definition of an operator isan entity that turns one function into another

H

/d dz


The particular operator is called the Hamiltonian operator

Just like the classical Hamiltonian functionit is related to the total energy of the system

This Hamiltonian idea extends beyond the specific Schrödinger-equation definition we have so far

which is for single, non-magnetic particlesIn general, in non-relativistic quantum mechanics

the Hamiltonian is the operator related to the total energy of the system

H

Operators and expectation values

Now we show a simple, important and general relation between

the Hamiltonian operatorthe wavefunction, and

the expectation value of the energyTo do so

we start by looking at the integral

where is the wavefunction of some system of interest

3ˆ, ,I t H t d r r r

,t r


In looking at this integral

we will expand the wavefunction in the (normalized) energy eigenstates

So

3ˆ, ,I t H t d r r r

,t r

, n nn

t c t r r

2

2ˆ , , ,2

H t V t tm

r r r

n r

2

2 ,2 n n

nV t c t

m

r r n n nn

c t E r


So the integral becomes

Because of the orthonormality of the basis functionsthe only terms in the double sum that survive

are the ones for which

so

But this is just the expectation value of the energy, so

3 3ˆ, , m m n n nm n

t H t d c t c t E d

r r r r r r

n r

n m

23ˆ, , n nn

t H t d E c r r r

3ˆ, ,E t H t d r r r

Benefit of the use of operators

Question:if we already knew how to calculate

from

why use the new relation?

Answer:We do not have to solve for the eigenfunctions of the operator to get the result

E2

n n n nn n

E E P E c

3ˆ, ,E t H t d r r r


Slides: Video 4.3.5 Time evolution and the Hamiltonian


Section 3.11


Time evolution and the Hamiltonian



Taking Schrödinger’s time dependent equation

and rewriting it asand presuming does not depend

explicitly on time i.e., the potential is constant

could we somehow legally write

,ˆ ,t

H t it

r

r

ˆ,

,t iH t

t

r

rH

V r

1 01 0

ˆ, exp ,

iH t tt t

r r


Certainly, if the Hamiltonian operator here was replaced by a constant number

we could perform such an integration of

to get

1 01 0

ˆ, exp ,

iH t tt t

r r

ˆ,

,t iH t

t

r

r

H


If, with some careful definition, it was legal to do thisthen we would have an operator that

gives us the state at time t1 directly from that at time t0To think about this “legality”

first we note that, because is a linear operatorfor any number a

Since this works for any functionwe can write as a shorthand

H

ˆ ˆ, ,H a t aH t r r

,t r

ˆ ˆHa aH


Next we have to define what we mean by an operator raised to a power

By we meanSpecifically, for example, for the energy eigenfunction

We can proceed inductively to define all higher powers

which will give, for the an energy eigenfunction

2H 2ˆ ˆ ˆ, ,H t H H t r r n r

2 2ˆ ˆ ˆ ˆ ˆn n n n n n n nH H H H E E H E r r r r r

1ˆ ˆ ˆm mH H H

ˆ m mn n nH E r r


Now let us look at the time evolution of some wavefunction between times t0 and t1

Suppose the wavefunction at time t0 iswhich we expand in the energy eigenfunctions

as Then we know

multiplying by the complex exponential factors for the time-evolution of each basis function

,t r r

n r n n

n

a r r

1 01, exp n

n nn

iE t tt a

r r


In

noting that

we can write the exponentials as power seriesso

1 01, exp n

n nn

iE t tt a

r r

2 3

exp 12! 3!x xx x

2

1 0 1 01

1, 12!

n nn n

n

iE t t iE t tt a

r r


In

because we showed thatwe can substitute to obtain

2

1 0 1 01

1, 12!

n nn n

n

iE t t iE t tt a

r r

ˆ m mn n nH E r r

2

1 0 1 01

ˆ ˆ1, 12!n n

n

iH t t iH t tt a

r r


With

because the operator and all its powers commute with scalar quantities (numbers) we can rewrite

H

2

1 0 1 01

ˆ ˆ1, 12!n n

n

iH t t iH t tt a

r r

2

1 0 1 01

2

1 0 1 00

ˆ ˆ1, 12!

ˆ ˆ11 ,2!

n nn

iH t t iH t tt a

iH t t iH t tt

r r

r


So, provided we define the exponential of the operator in terms of a power series, i.e.,

then we can write our preceding expression as

2

1 0 1 0 1 0ˆ ˆ ˆ1exp 1

2!iH t t iH t t iH t t

1 01 0

ˆ, exp ,

iH t tt t

r r


Hence we have established that there is a well-defined operator that

given the quantum mechanical wavefunction or “state” at time t0

will tell us what the state is at a time t1

1 01 0

ˆ, exp ,

iH t tt t

r r

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