Understanding Mathematics of Quantum Physics

[mpsychdoctor]

Hello all. I'm new to the forum, and I'm not a physicist. I'm a psychologist. I need some help understanding the meaning of the mathematics applied in quantum physics, when calculating the probability for a particle being located at a certain point in space and time. I understand that there are two wave functions calculated to indentify the potential: a real wave function representing a forward-in-time wave and an imaginary wave function representing a backward-in-time wave. Is that right? I understand that the referent of the backward-in-time wave is the particular point in space and time where the particle potentially may be found, and that many such points mean that there are many referents for the imaginary wave function. Is that right? What I don't understand is the referent of the forward-in-time wave function. Is the referent of the real wave function held constant for every possible point in space and time where that particle may be found? Please help. Or, direct me to where I can get help. Thanks.

 

[selfAdjoint]

I think you have picked up some misinformation somewhere.

The values (called amplitudes) of both the forward in time wave and the backward in time companion are complex numbers, and the two of them are complex conjugates. If $$\psi = a + bi$$ is any complex number, $$\psi^* = a - bi$$ is its complex conjugate. The product of the two is $$\psi^* \psi = a^2 + b^2$$, a real number. Physicists then interpret this real number as the probability of finding the particle at the point where these particular values of the wave function appear.

It's fun to think of $$\psi^*$$ traveling back in time, and there is a technical sense in which this is a meaningful statement. But it is dangerous for people without that technical training to interpret such a statement or draw conclusions from it. I am sure as a psychologist you are familiar with material in your own area that has that same danger for the non technically trained.

 

[marcus]

Originally posted by mpsychdoctor
Hello all. I'm new to the forum, and I'm not a physicist. I'm a psychologist. I need some help understanding the meaning of the mathematics applied in quantum physics,...

join the club. IMHO (in my humble opinion) we all need help
understanding that

selfAdjoint, who has the wisdom of long experience, has mentioned "danger" and classified people into the safe (techn. trained) and the unsafe (not techn. trained)
but let us consider what the danger is.
Isnt it merely the risk of hopeless confusion and horrendous frustration and (what is more to the point) becoming tiresome to others?
A layman confused by quantum mechanics is a ghastly sight which can cause even a brave physicist to break out in cold sweat.

But for the layman, who does not realize how monstrous his misconceptions are, the experience of confusion may not be so bad!
The danger to him personally is not prohibitive. And should not deter him.

So let us grant Mpsycho permission to recklessly wonder about "the meaning of mathematics applied to quantum physics".


As basis for discussion here are some (possibly provocative and oversimplified) assertions. The key element in every quantum theory is a Hilbertspace. A specially nice type of vectorspace. what we know and don't know is represented in that hilbertspace. In any quantum theory, observations are represented as linear OPERATORS in the hilberspace belonging to the theory.

the "wave functions" are somebody's semimythical visualizations of elements of a certain hilbertspace. what they look like does not matter and the stories people tell about them do not matter.
to understand "the meaning of mathematics applied to q. physics" you should first of all focus attention on what is at the heart of every quantum theory. Understand the logic of the operators on a vectorspace and how they can represent information and ignorance.
this is the hard core of q. theory. Please correct me if I am mistaken about this!

Also bear in mind the ARBITRARINESS of q. theory. No one on the planet has yet explained why it works so well to represent the states of knowledge, uncertainty, and observation in this essentially geometrical lattice of subspaces of an abstract vectorspace. Someone said "let's try this" and it worked. A coherent creation story of why the world should (approximately at least) work that way has never, AFAIK, been told.

After all, hilbert spaces were only invented in the early XXth century by a man who lived in Goettingen named David Hilbert. Why in god's name should this contraption be at the core of QM

So there is an element of irrationality in q. theory that is almost taboo to discuss. Outspoken taboo-breakers like Feynman make funny jokes like: "Anyone who believes they understand quantum mechanics has not been paying attention!" Who was it actually that said that?

the essential lay/clerical dividing line is this: the trained (in other words, "safe") initiates have GOTTEN OVER the irrationality. It disturbed and frustrated each of them at some time in the past and they got over it and learned to calculate the right numbers. until the pits of doubt in their minds have been paved over with habit

it strikes me that a psychologist who asks about the intellectual foundations of quantum theory is showing some originality and might actually be amused if he could be told. if you are willing to try, selfAdjoint, I will help. but if not I will leave off

 

[mpsychdoctor]

Great! So, what can you tell me about "the logic of operators in vector space"?

I will not ignore the best representation of my reality that humans have ever discovered. Quantum mechanics is the best representation of my reality - why should I not seek to understand that? I have been grappling with this quantum mechanical quandary since I was 18 yrs. old. It is only now that I feel able to ask physicists semi-appropriate questions about the subject. I would like to know "how monstrous my misconceptions are". The "safe" may ask themselves if they are not themselves a victum of quantum mechanics. Selective exclusion of perceptions is what led physicists to believe that a photon was a particle. Self-hypnotized, they suspend doubt, and enter a state of flow - practicing science. But if I am correct, this is necessary when the real wave function is applied to human functioning. I mean no disrespect to Self-Adjoint; it is inherent to human functioning that we selectively exclude certain perceptions. But I have chosen not to exclude the elements of quantum mechanics. I understand that, whatever conclusions I might ultimately draw about the meaning of the mathematics applied in quantum mechanics, that my conclusions will be incomplete.

Wave functions are visual tools used to represent the elements in hilbert space; these elements are linear operators. There are two elements in hilbert space, a real element and an imaginary element. These two elements can be best visualized as 2 waves: all known and unknown information (regarding the outcome of a particular experiment). The fact that they are linear operators adds the time dimension, I believe? So, the questions are: what is the sphere of the known and the sphere of the unknown, when visualized along the time dimension represented by the linear operator? To understand this, I would suggest examining the key element of the "observer effect".

As a psychologist, I find it presumptuous to limit the field of observation to mere "measuring apparatus". The observer effect is not necessarily limited to the technological measure. I believe all aspects of the experimental methodolgy need to be considered when attempting to understand the observer effect. From hypothesis formation to measuring technology, and also the experimental outcome; each of these may be significant to the observer effect.

So, what I understand is that the linear operator's time dimension's have two spheres (points) represented upon an axis of time. The simple logic suggests the same. The known information/sphere is the present because if it is not known in the present it is unknown, and therefore it is in the opposite sphere of unknown information. This necessarily implies a time dimension: all unknown information is potentially known information: can be known in the 'future'. Hence, the imaginary wave function backward in time. These are the two spheres. In the scientific method, these two spheres are clearly represented. A scientific experiment can be divided into known and unknown divisions; the dividing line of awareness is the point of human observation, typically an observation provided by some technological measure.

This may be significant: The scientist's choice's during the development of his/her experimental methods has a significant impact upon the outcome of that scientist's experiment. Could the forward in time wave be the behavior of the scientist prior to the observation. Significantly, the scientist's experimental hypothesis will have a significant effect upon the scientist's choice of measure.

The choice of measure is agreed by most to be significant to this quantum quanrary. This choice alters the potentials in the sphere of the unknown: certain information is necessarily included by the chosen measure, but other information is excluded. The other information will not be observed by the scientist, because she/he is looking in the wrong direction to see that information. The chosen measure can't detect that information. In a sense, the scientist has altered the potentials along the 'possibility wave'. During the experiment, at the point of observation (point where the chosen measure is applied), the wave function collapses. The collapse defines the dividing line between the known and the unknown, between the present and future.

The two linear operators are represented as waves. When considering the time element, it may not be too far fetched to consider these waves flowing in time. Hence, waves forward and backward in time. The sphere of unknown information is appropriately termed 'imaginary'. The experimental hypothesis is an imaginary outcome. All unknown outcomes for a particular experiment are imaginary outcomes. Only the past and the present can be known, the future is unknown. I know this may be hard to hear, but I think this implies that the scientist's experimental hypothesis exists in some fantasy world. Do not lose heart! The good part is that quantum mechanics strongly suggests that the imaginary realm must be considered as somehow real.

Is this a tenable interpretation of the data? Can the referent of the 'real wave function' be the experimental scientist at the moment of measured observation? Tell me about the logic of operators in vector space.

 

[selfAdjoint]

Let me go back and show you an example, the position of a particle on a line. I'll try to avoid notation.

To find out where a particle is on a given line (assuming it's physically constrained to move on that line), you have to make a measurement. Corresponding to the measurement there will be a linear operator on the Hilbert space of the particle. The "points" of the Hilbert space are vectors and the operator is a linear transformation that maps each vector into some other vector, the way functions do, but it's linear so it maps the sum of two vectors into the sum of their images and so on. So the linear transformation takes the values of the wave function (complex vectors) and maps them into some other complex vectors.

Now a new concept: eigenvalues. It can happen that for _some_ vectors, the linear transformation just maps them into a multiple of themselves. I need a little notation here. $$T(v) = \alpha v$$. The action of T on this particular v is just to multiply it by $$\alpha$$. And this multiplier is called an eigenvalue of T, and v is called an eigenvector of T.

Now any transformation might have eigenvalues - there's a big theory of transformations and eigenvalues - but some transformations are even more special; they have REAL NUMBER eigenvalues, even though all the vectors they act on are complex-valued. Such transformations are called Hermitian (and sometimes self Adjoint, heh). And here is the bottom line: The position operator is Hermitian! Its eigenvalues are real numbers, not complex numbers!

So you do the experiment, and the position operator acts on the complex-number values of the wave function in the Hilbert space, and the EIGENVECTORS correspond to the coordinates which you might find for the particle on the line, and the REAL EIGENVALUES give the probabilities that you will find it at each of the possible coordinates.

Notice the non-classical behavior here. You don't get a definite position out of your measurement, you get a bunch of possible positions and probabilities for each one to be the true one.

 

[jcsd]

I've just noticed this thread, you shouldn't really cross post mpsychdoctor. Anyway you'll notice that in different ways me and self-adjoint havepretty much said the same thing, accept that he's defined the probailty of finding a particle at a certain poitn as the product of the wavefunction and it's complex conjugate, whereas I defined it as the square modulus, but you should also see that they are infact both equal to each other.

i.e.

$$z\bar{z} = |z|^2$$

In the transactional interpetaion the complex conjugate of the Schroedinger equation is considered to eb a backwards in time advanced wave.