Reproducibility of measurements in quantum mechanics

[Geofleur]

There is a statement in the Landau and Lifgarbagez volume on quantum mechanics that has had me puzzled for weeks now. They say (bottom of pg. 23):

"If the electron was in the state psi_n [an eigenstate of f], then a measurement of the quantity f carred out on it leads with certainty to the value f_n. After the measurement, however, the electron is in a state phi_n different from its initial one, and in this state f does not in general take any definite value."

They seem to be saying that if you make a measurement and get f_n and then repeat the measurement immediately, you will not in general get f_n again! But the relevant experiments I know of do not seem to display such behavior. In the Stern-Gerlach experiment, for example, if a beam of electrons is determined to be spin up and then passed through the apparatus again, all the electrons in the beam will continue to be spin up. What am I missing here?

 

[Simon Bridge]

$\renewcommand{\ket}[1]{\left | #1 \right \rangle}$

"If the electron was in the state psi_n [an eigenstate of f], then a measurement of the quantity f carred out on it leads with certainty to the value f_n. After the measurement, however, the electron is in a state phi_n different from its initial one, and in this state f does not in general take any definite value."

That doesn't look right does it?

If $\small \psi_n$ is the nth eigenstate of $\small \hat{A}$ and the system is prepared in that state, then all measurements of observable $\small A$ should produce $\small a_n$.

It's not clear from the passage what they are talking about - what is the chapter about?

 

[Simon Bridge]

That would be this one?
http://archive.org/details/QuantumMechanics_104

... the chapter is about basic concepts in QM

The top of page 24 explains that the states $\Psi_n(q)$ do not have to correspond with the states $\phi_n(q)$ - the latter do not have to be orthogonal, nor be eigenstates of an observable.

 

[bhobba]

"If the electron was in the state psi_n [an eigenstate of f], then a measurement of the quantity f carred out on it leads with certainty to the value f_n. After the measurement, however, the electron is in a state phi_n different from its initial one, and in this state f does not in general take any definite value."

Unless clarified in some way by context its not correct. Continuity implies immediately after an observation it must be in the corresponding eigenstate of the eigenvalue that was the result of the observation - see Chapter 8 of QM - A Modern Development by Ballentine where this is proven.

Despite the very high esteem I hold Landau in and how much I love his first book in that series on Mechanics I think its best, considering the progress that has been made in QM since Landau's time, to study it from a more modern text like Ballentine. Here it is developed in a fairly rigorous way from two axioms and stuff like the above derived rather than assumed.

Thanks
Bill

 

[Simon Bridge]

I'm with bhobba on this one - Landau is making something of a meal of the measuring apparatus operation in the quantum measurement. I think the wording will be very misleading to a modern student who has seen similar notation used for slightly different concepts.

The main advantage of these older texts is that they are available free-of-charge online.

 

[Haborix]

Most everything has been said, but I would like to try and lay out what I think Landau is trying to argue.

The initial assumption is that upon measurement the system of electron and apparatus is projected into a "classical" eigenfunction (stated in the paragraph between equations 7.2 and 7.3). This classical eigenfunction contains normalized wavefunctions $\small \phi_n$ weighted by the $\small a_n$. It is the $\small a_n$ that are related to the $\small \Psi_n$. This is the contradictory part (conflicting with his initial assumption), $\small \Psi_n$ are the orthonormal eigenfunctions that describe the electron pre-measurement (These will be the wavefunctions that are the eigenfunctions of your observables). What he's trying to sell is that the classical apparatus stamps this new $\small \phi_n$ on the electron. So once you measure the first time you may be able to glean some value for an observable $\small f_n$, but the new state after measurement is $\small \phi_n$ which, as he states, may not be equivalent to $\small \Psi_n$, and thus you aren't guaranteed to get the value $\small f_n$ for your observable upon additional measurements.

It's a little wonky.

 

[vanhees71]

Ironically Landau/Lifhitz, being pretty old fashioned with their overemphasizing of wave mechanics and the lack of the representation free bra-ket notation, is one of the few books that get this point on measurement theory right. The collapse assumption is for all practical purposes (FOPP) correct for and only for ideal von Neumann filter measurements. This is a very special case of a measurement, which I'd rather call a state preparation than measurement for clarity.

This is nicely explained in modern notation in Ballentine, p. 233. It's also worth reading the next section. It's the clearest and shortest description of the meaning of pure states, I've ever seen:

(a) A pure state $|\psi \rangle$ [here, I'd be more rigorous and define a pure state either by the ray or the statistical operator $\hat{R}_{\psi}=|\psi \rangle \langle \psi|$] provides a complete and exhaustive description of an individual system. A dynamical variable represented by the operator $\hat{Q}$ has a value ($q$, say) if and only if $\hat{Q}|\psi \rangle=q|\psi \rangle.$

(b) A pure state describes the statistical properties of an ensemble of similarly prepared systems.

 

[bhobba]

This is nicely explained in modern notation in Ballentine, p. 233. It's also worth reading the next section. It's the clearest and shortest description of the meaning of pure states, I've ever seen:

(a) A pure state $|\psi \rangle$ [here, I'd be more rigorous and define a pure state either by the ray or the statistical operator $\hat{R}_{\psi}=|\psi \rangle \langle \psi|$] provides a complete and exhaustive description of an individual system. A dynamical variable represented by the operator $\hat{Q}$ has a value ($q$, say) if and only if $\hat{Q}|\psi \rangle=q|\psi \rangle.$

(b) A pure state describes the statistical properties of an ensemble of similarly prepared systems.

Indeed.

The clarity of Ballentine in explaining such issues is IMHO what lifts it above other QM books I have read and/or own, and why it is always my go-to book. He develops it from two axioms - they are basically the only two assumptions that need to made - the rest follows from things like continuity and the invarience of probabilities.

I simply cannot recommend it highly enough to anyone that wants to learn QM. I well remember when I first studied that book many moons ago now. I had read a few books like Dirac's and Von Neumann's and was dissatisfied with both for various reasons - but with Ballentine everything became sweetness and light. Suffice to say it had a big impact on me.

Thanks
Bill

 

[Geofleur]

Thanks everyone, this helps!