In Fredrik's recent thread:(adsbygoogle = window.adsbygoogle || []).push({});

https://www.physicsforums.com/showthread.php?t=506326

some questions arose about this paper:

J. B. Hartle, "Quantum Mechanics of Individual Systems",

Am. J. Phys., vol 36, no 8, (1968), pp704-712.

[For other readers, you'll need to have read or be able to access a

copy of Hartle's paper above for what follows to make most sense.]

In that paper, Hartle claims to derive statistical aspects of QM (aka

Born rule) from the nonstatistical aspects of QM. Fredrik was skeptical

about whether Hartle's paper really does derive the Born rule, and had

various other doubts about the content of the paper. Since this is all a

bit tangential to the main topic of his thread, I post my attempt at a

more detailed explanation here as a separate thread.

Let A be an observable with eigenvalues and eigenvectors as follows:

[tex]

\def\<{\langle}

\def\>{\rangle}

\def\half{\frac{1}{2}}

\def\third{\frac{1}{3}}

A |i\> = a_i |i\>

[/tex]

In QM, one asserts that the value of the observable A measured in state

[itex]|i\>[/itex] is [itex]a_i[/itex] and that this is definite (i.e., deterministic).

This is called a "nonstatistical" assertion of QM about an individual system

modeled by [itex]|i\>[/itex].

Hartle's assertion (p706, lower right) is that the statistical predictions

of QM can be recovered from QM'snonstatisticalassertions about

individual systems. His argument involves constructing an N-fold tensor

product space and depends on a particular interpretation of the following

operator acting on that space [see Hartle's eq(5)]:

[tex]

f_N^k ~:= \sum_{i_1,\cdots,i_N} |i_1,1\> \cdots |i_N,N\>

\left( \frac{1}{N} \sum_{\alpha=1}^N \delta_{k i_\alpha} \right)

\<i_N,N| \cdots \<i_1,1|

[/tex]

which Hartle calls thefrequency operator. The sum over each

[itex]i[/itex] parameter means we are summing over the eigenvectors of A

in each (mutually independent) component space of the N-fold tensor

product space. The component spaces are used to model independent

repetitions of the experiment (such that the prepared state of the system

may or may not all be the same).

To explain why [itex]f_N^k[/itex] deserves the name "frequency operator"

we consider some specific cases. For the trivial case N=1, the frequency

operator reduces to

[tex]

f_1^k ~=~ \sum_{i_1} |i_1,1\> \left( \delta_{k i_1} \right) \<i_1,1|

[/tex]

or, dropping the component space label (which is unnecessary when N=1),

[tex]

f_1^k ~=~ \sum_{i} |i\> \delta_{ki} \<i| ~=~ |k\>\<k|

[/tex]

For N=2, the frequency operator is

[tex]

\begin{align}

f_2^k

&= \sum_{i_1,i_2} |i_1,1\> |i_2,2\>

\Big( \half \sum_{\alpha=1}^2 \delta_{k i_\alpha} \Big)

\<i_2,2| \<i_1,1| \\

&= \sum_{i_1,i_2} |i_1,1\> |i_2,2\>

\half \Big( \delta_{k i_1} + \delta_{k i_2} \Big)

\<i_2,2| \<i_1,1| \\

&= \half\Big(

\sum_{i_1,i_2} |i_1,1\> |i_2,2\> \delta_{k i_1} \<i_2,2| \<i_1,1|

~+~ \sum_{i_1,i_2} |i_1,1\> |i_2,2\> \delta_{k i_2} \<i_2,2| \<i_1,1|

\Big) \\

&= \half\Big(

\sum_{i_2} |k,1\> |i_2,2\> \<i_2,2| \<k,1|

~+~ \sum_{i_1} |i_1,1\> |k,2\> \<k,2| \<i_1,1| \Big) \\

&= \half\Big( |k,1\> \<k,1| ~+~ |k,2\> \<k,2| \Big)

\end{align}

[/tex]

where we have used the mathematical theorem that the eigenvectors of

the self-adjoint operator A give a resolution of unity. Note that each term in

the last line above includes an implicit identity operator on the other

component space of the 2-fold tensor product, e.g.,

[tex]

|k,1\> \<k,1|

[/tex]

acts as a projector on component space 1, but as the identity on

component space 2.

Similarly, for N=3 we find

[tex]

f_3^k

~=~ \third\Big( |k,1\> \<k,1| ~+~ |k,2\> \<k,2| ~+~ |k,3\> \<k,3| \Big)

[/tex]

and so on. Perhaps it would have been clearer to define the

frequency operator as

[tex]

f_N^k

~=~ \frac{1}{N} \sum_{\alpha=1}^N |k,\alpha\> \<k,\alpha|

[/tex]

Consider now how [itex]f_3^k[/itex] acts on three independent experiments,

with preparations corresponding to definite outcomes, i.e., eigenstates of A.

If the sequential preparations are, respectively, [itex]|j,1\>[/itex], [itex]|k,2\>[/itex], [itex]|k,3\>[/itex] (with [itex]j\ne k[/itex]),

then [itex]f_3^k[/itex] acts on the tensor product of these states as follows:

[tex]

\begin{align}

f_3^k ~ |j,1\> |k,2\> |k,3\>

&= \third\Big( |k,1\> \<k,1| ~+~ |k,2\> \<k,2| ~+~ |k,3\> \<k,3| \Big)

|j,1\> |k,2\> |k,3\> \\

&= \frac{2}{3} \; |j,1\> |k,2\> |k,3\>

\end{align}

[/tex]

giving the correct answer that [itex](2/3)[/itex] of the initial preparations were in state [itex]|k\>[/itex].

This is the justification for calling [itex]f_N^k[/itex] a "frequency

operator" -- in the following sense. Hartle takes the nonstatistical

aspects of QM as given. Initial states prepared in an eigenstate of A

fall into this category: the result of measuring A for any of these

states is definite (deterministic). [This is analogous to a

sequence of trivial classical experiments in which a state is prepared

(e.g., a coin laid flat on a table by Ian) and then examined (without

tossing the coin) by Fred. Fred always observes precisely whatever state

Ian prepared.]

Now we investigate the behaviour of [itex]f_N^k[/itex] applied to the

product of N copies of anarbitrarystate [itex]|s\>[/itex] which is

in general a linear combination of the [itex]|i\>[/itex] eigenstates of

A. We denote N-fold tensor product state in an abbreviated form:

[tex]

|(s)^N\> ~:=~ |s,1\> |s,2\> \cdots |s,N\>

[/tex]

Hartle shows that

[tex]

\lim_{N\to\infty} \Big( f_N^k - |\<k|s\>|^2 \Big) |(s)^N\> ~=~ 0

[/tex]

In other words we may say that, [itex]|(s)^\infty\>[/itex] is an eigenstate of [itex]f_\infty^k[/itex] with eigenvalue [itex]|\<k|s\>|^2[/itex].

Relying again on the nonstatistical aspects of QM, this means that if we measure

the observable [itex]f_\infty^k[/itex] for the state [itex]|(s)^\infty\>[/itex] we definitely get [itex]|\<k|s\>|^2[/itex].

All that now remains is the question of whether it's still reasonable to call

[itex]f_\infty^k[/itex] a "frequency" operator when applied to states that are not

eigenstates of A.

First, consider other well-known observables in QM, such as momentum or spin.

We construct the associated operators so that they have the physically correct

spectrum of eigenvalues. (Operationally, this corresponds to the set of outcomes

possible from an apparatus which "measures" that observable, which has itself been

designed and calibrated by preparing certain states and verifying that the apparatus

always gives the same answer whenever presented with that particular prepared state).

In other words, the candidate operator must give the correct answer in every classical

(deterministic) case.

But that's exactly what Hartle did in defining his frequency operator. He constructed

it so that it gave the physically correct answers when applied to its (finite-N) eigenstates.

So the answer to the question of why it's reasonable to say that [itex]f_N^k[/itex] is a

"frequency" operator is essentially the same as the answer to why it's reasonable to

consider [itex]i\partial_x[/itex] a "momentum" operator when acting on a space of

functions: it has the physically sensible set of eigenvalues, as verified by its action on

states which we expect physically should be its eigenstates, corresponding to

deterministic classical cases.

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# Hartle: QM of Individual Systems (1968)

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