# Experimental Quantum Mechanics Texts?

1. Oct 19, 2014

### kmm

In studying QM, it's been very abstract. I'll read things such as, we get an eigenvalue when we measure an observable. Or we prepare an ensemble of particles in the state $\Psi$. I understand this in an abstract way, but I don't understand what it actually means to get an eigenvalue when a measurement is made. How is an eigenvalue "observed" in the laboratory? This is just one example but I'm looking for a textbook that ties these sorts of things in Quantum theory to the experimental side of things. Not necessarily a textbook but any source. Any suggestions? Thanks

2. Oct 19, 2014

### Staff: Mentor

Its not an experimental matter - its got to do with the foundational principles of QM which many different experiments, and much thought about their implications, led to.

Conceptually its tied up the necessity of continuous transformations between pure states to model physical systems.

Suppose we have a system in 2 states represented by the vectors [0,1] and [1,0]. These states are called pure. These can be randomly presented for observation and you get the vector [p1, p2] where p1 and p2 give the probabilities of observing the pure state. Such states are called mixed. Now consider the matrix A that say after 1 second transforms one pure state to another with rows [0, 1] and [1, 0]. But what happens when A is applied for half a second. Well that would be a matrix U^2 = A. You can work this out and low and behold U is complex. Apply it to a pure state and you get a complex vector. This is something new. Its not a mixed state - but you are forced to it if you want continuous transformations between pure states.

QM is basically the theory that makes sense out of pure states that are complex numbers. It is this that allows for interference effects - these cant be explained via classical probability theory.

A more detailed account of that approach can be found here:
http://arxiv.org/pdf/quantph/0101012.pdf

If you want to see an axiomatic approach from a single axiom see post 137:

The following from the above link explains how eigenvalues come into it:
'To derive Ballentine's two axioms we need to define what is called a resolution of the identity which is POVM that is disjoint. Such are called Von Neumann observations. We know from the Spectral theorem Hermitian operators, H, can be uniquely decomposed into resolutions of the idenity H = ∑ yi Ei. So what we do is given any observation based on a resolution of the identity Ei we can associate a real number yi with each outcome and uniquely define a Hermitian operator O = ∑ yi Ei, called the observable of the observation.'

The key thing is the mapping of outcomes to resolutions of the identity. Eigenvalues come into it via the Spectral Theorem.

Thanks
Bill

Last edited: Oct 19, 2014
3. Oct 19, 2014

### vanhees71

Well, that's all right what bhobba writes, but physics is after all an empirical science. Everything in theory is about the prediction of the outcome of observations and measurements, which are observations in a precise quantified form. Of course, we do not observe rays in a Hilbert space and eigenvalues of self-adjoint operators when we observe particles.

To the contrary states in the sense of quantum theory should not be understood as such abstract concepts but as (equivalence classes) of preparations of real systems. E.g., if you want to measure the cross section of a certain scattering process, it's not defined by a transition probability (S-matrix element squared) but by an experimental setup as is provided by accelerators in the labs around the world: You prepare a beam of particles with quite well defined energy and momentum of the particles making up these beams. These beams of particles hit a target or another beam of particles (if you have a collider as the LHC an CERN) and then a detector (which nowadays usually is a clever collection of many subdetectors) registers the particles coming out of such collisions. Of course, you detect a lot of such always (as good as technically possible) equally prepared particle beams, and then analyze the outcome of the detectors with well-defined statistical methods, estimate systematic errors and all that.

It would be very wrong to describe quantum physics only as an abstract theoretical edifice and forgetting the reality provided by experimental setups. It appears to me as that is quite often forgotten in the theoretical community, which sometimes even tends to drift towards quite contraproductive metaphysical discussions about what "reality" or "realistic interpretations of quantum theory" might be or whether this is possible. These are mostly quite empty and useless debates. Last but not least "reality" must be always defined as something an experimentalist can handle in the lab, precisely defining the conditions under which the objects of interest are prepared and how the outcome of measurements are operationally defined. Then many apparent "problems" of quantum theory resolve by themselves.

4. Oct 19, 2014

### atyy

A very simple example of an eigenvalue measurement is the detection of a particle in a double slit experiment. There the measured (transverse) position is a position eigenvalue. A second simple example of an eigenvalue measurement is that in a double slit experiment, if the screen is placed at infinity, the measurement of (transverse) position at infinity plus some calculation measures the (transverse) momentum of the particle just after the slit. This is because momentum is the Fourier transform of position, and wave function evolution in the large distance Fraunhofer limit yields the Fourier transform of the initial wave function. http://www.atomwave.org/rmparticle/ao%20refs/aifm%20pdfs%20by%20group%20leaders/pfau%20pdfs/PK97.pdf [Broken].

In a Stern-Gerlach experiment, the measurement of the spin eigenvalue is indirect. What is directly measured is the position of the particle, from which the spin is measured. http://www.nucleares.unam.mx/elaf2013/notes/Mello_ELAF_2013/Mello_ELAF_2013.pdf [Broken]

In fact, most measurements are indirect. In most measurements, an ancilla interacts with the particle and becomes entangled with the particle. Then the ancilla is measured, and the particle is indirectly measured because it has become entangled with the ancilla in the course of the measurement. http://www-bcf.usc.edu/~tbrun/Course/lecture08.pdf
http://arxiv.org/abs/1110.6815

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5. Oct 19, 2014

### vanhees71

By the way, does anybody know indeed a good experimental textbook which describes quantum theoretical measurements from a really experimental point of view?

What's very important in conncection with the above given example of position and momentum measurements is that in fact there are no eigenvalues of positions and momenta. It's only spectral values in the continuous spectrum of the associated self-adjoint operators. That's indeed reflected by the experimental fact that you always must use a slit of finite extension. There is no such thing as an "infintesimally small" slit that would determine a precise position of the particle. Of course, this is also reflected in the quantum-theoretical formalism by the fact that there is no eigenstate of the position or momentum operator. The corresponding Dirac-$\delta$ or plane-wave solution of the position-picture eigenvalue problems of the operators are generalized functions (distributions) not square-integrable wave functions.

6. Oct 19, 2014

### atyy

In theory a completely precise (non-relativistic) position measurement is possible. http://arxiv.org/abs/0706.3526 (section 2.3.2) For continuous variables the projection postulate has to be generalized, and the state of the particle after the measurement depends on the initial state of the ancilla.

However, a description of a position measurement with finite resolution, which is closer to real experiments, is given by http://arxiv.org/abs/1211.4169.

Also http://scitation.aip.org/content/aapt/journal/ajp/62/11/10.1119/1.17657 (but not free).

Last edited: Oct 19, 2014
7. Oct 19, 2014

### kmm

Thanks for the input and reading material everyone. This gives me a lot to ponder on.

Yes, it would still be nice to know of any books like this.

8. Oct 20, 2014

### Demystifier

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9. Nov 25, 2014

### luxor99

Beck's book is the one to get. He starts with polarization (with detailed experiments that can be done in the lab) as a gentle 2D vector introduction to QM. Most other texts start with spin (which is abstract unless you're familiar with Stern-Gerlach experiment), bore you with the formalism, or even worse, jump right into wave mechanics and the Schrodinger equation.

By the way, if anyone has the solutions manual for this book, please ping me.

10. Nov 26, 2014

### vanhees71

I heavily disagree with the statement that an approach starting with massive spin-1/2 particles is worse than starting with polarization (of light, I guess). On a superficial view both are pretty much the same, and this is true to some extent concerning the math as long as you close you eyes to the detail that photons are described as relativistic massless spin-1 fields. To begin with this case, is didactically very questionable. Usually you beginn with the most simple case possible, and that's a massive particle in situations where non-relativistic quantum theory is applicable. When I had to teach the quantum theory 1 lecture, I'd start with exactly the example of spin 1/2 and the Stern-Gerlach experiment, which can be presented as a correct example from the very beginning. Starting with photons almost certainly implies wrong ideas about them in the heads of quantum beginners, which then have to be eliminated again later, when one teaches relativistic QFT.

11. Dec 9, 2014

### Demystifier

Now I have seen the book, and I can tell it's great. It is a comprehensive pedagogically written text containing both basic and advanced topics in quantum physics. However, even though the author is an experimentalist, I am not convinced that experimentalists will particularly like it. I think it is much better suited to practical theoreticians than to experimentalists.

Last edited by a moderator: May 7, 2017