What is the meaning of the 'real wave function' in quantum mechanics?

[mpsychdoctor]

Hello! I need someone to assist me in understanding some of the calculations involved in quantum mechanics. Specifically, I have questions about the "real wave function" and the "imaginary wave function". I understand that the 'imaginary wave function' is a backward-in-time function and that the 'real wave function' is a forward-in-time function. I also understand that the 'imaginary wave function' refers to a possible experimental outcome in Hilbert space. What does the 'real wave function' refer to? Is there a spatiotemporal point it refers to? Thank you.

 

[PrudensOptimus]

post some sample problems and see.

 

[mpsychdoctor]

Math is not my specialty

The problem is, I don't understand things mathematically. So, I don't have any sample problems to post. I'm trying to understand the general point in space-time from where the 'real wave function' is calculated 'forward-in-time' from. My sample problem is this: Backward-in-time from one of an infinite number of experimental outcomes and toward _____ (?), and forward-in-time from ______ (?) toward one of an infinite number of possible experimental outcomes. If I'm not placing the question in a form that can be answered, give me an example of an appropriate sample problem. But, please, try to translate your sample problem into a verbal form. Thanks.

 

[jcsd]

I think your trying to run before you can walk you seem to be slightly confused with advanced waves, etc.

This is the (one-dimensional time dependent) Schroedinger equation:
$$i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi$$

Ψ is the wavefunction and it may be imaginary

In th convential intepretation of quantum mechanics, the proabilty of finding a particle at a certain point at a certain time is represnted as follows:

$$P(x,t) = |\Psi(x,t)|^2$$

That is the probability of finding a particle at a certain point at a certain time is equal to the square modulus of the (normalized) wavefunction. It's clear that while the wavefunction may be imaginary the square modulus of the wavefunction may not.

 

[PrudensOptimus]

lol that differential equation would scare most people away.

give some easy examples lloll

 

[mpsychdoctor]

eeks!

Ok. I know you are trying to help, but you're not understanding how ignorant I am. I understand about the probability of a particle being at a certain point in space and time. What I don't understand is the forward-in-time wave function, the real wave function. Let me put it another way: To calculate the probability that a particle will be at a certain point in space and time - that's one calculation. Is a second calculation necessary for identifying the probability that the particle will be in a second/other point in space and time? And so on for a third point in space and time, and the forth. Are there separate calculations for each point in space and time? My understanding is that there are separate calculations for each potential. Now, for each of those separate calculations, do they all have the same referent in the present? That is, although there are different possible futures for where that particle will be found, is there a single reference point from where that particle originated from? And, where/what is that starting referent point? What is the referent for the forward-in-time wave? If you don't know how to explain this to me (ignorant as I am), do you know who might be able to help me? Thanks for your patience.

 

[jcsd]

Basically that is all you need, the wavefunction is complex (that is it has imaginary and real parts), but the square modulus will give you the probailty of finding a particle at a certain poitn at a certain time. Classical wave equations are also complex but only the real part is given physical significance. It can also be written in a time independnt form which will give you the probailty of finding a particle at a certain point in space.

In the Copenhagen intepreation you don't have waves traveling back in time, so it's probably best to ignore the concept of adavnced waves for the moment.

 

[mpsychdoctor]

So, "its probably best to ignore the concept of advanced waves for the moment." But I would like an answer to my question! What do you mean, "for the moment"? You mention that the real wave function is a physical object in classical and quantum physics. So, what is the origin of the real wave? I am interested in understanding the referent point of the real wave function. I know what might help. Tell me, what are the mathematical symbols that represent the real wave function? The real wave symbols used in the formula? Then, tell me what those symbols represent. What do they stand for? What do those symbols mean? Tell me using a propositional format, the mathematics are beyond me.

Also, you mentioned using the copenhagen interpretation. I thought the Schrödinger mathematical solution was the same in both interpretations? Besides, the copenhagen interpretation ignores essential discoveries. It is clear that the experimenter's choice of measure has a significant effect upon the outcome of the experiment. The copenhagen interpretation ignores this fact, leaving other interpretations, such as the many worlds view, to be more tenable. Thanks for your time.

 

[ranyart]

Originally posted by mpsychdoctor
Hello! I need someone to assist me in understanding some of the calculations involved in quantum mechanics. Specifically, I have questions about the "real wave function" and the "imaginary wave function". I understand that the 'imaginary wave function' is a backward-in-time function and that the 'real wave function' is a forward-in-time function. I also understand that the 'imaginary wave function' refers to a possible experimental outcome in Hilbert space. What does the 'real wave function' refer to? Is there a spatiotemporal point it refers to? Thank you.

Ok the first thing to understand is what is Probable?..and what is not Probable.

Take any wavefunction that is propergating in the 'Now', its a realtime event function that is used in calculations. Now the imaginary wavefunction in whatever form you wish to use it(the backward-in-time-function) cannot, I repeat cannot exist? or give relative understanding.

And here is why, without giving the actual equations, I take it from your thread that you Understand that:

A)The Shroedinger Wavefunction relates to a 'now' tense, and the equation shows that it propergates away from source and continue's along a 'time-line' that is forever 'Now', the present wavefunction.

B)The Imaginary wavefunction relates to a 'NOT-NOW' context, here we are saying it in the terms of 'Backward-function', the 'Past'.

The first problem is that the present time SH-WF (Shroedinger wavefunction) is bound to the fact that the Universe is deemed to be in Expansion, one can derive this fact and connect the SH-WF by this virtue, and is linked to a mapping of the Universe expansion term, this is to say that it is the Universe's EXPANSION-TERM'S..little brother!

Now if one was to place this expansion SH-WF into and arena where the Universe is in a CONTRACTING-PHASE, then one can see that the present-time function(SH-WF) cannot and does not propergate, the wavefunction becomes 'Inverse', or Imaginary.

A good example is using the Imaginary Wavefunction and placing it in a Point in time of our past Universe time, say 300,000 years after the big-bang, the Universe is STILL an Expanding Universe and any inverse or 'collapsing-wavefunction' will not reveal anything except the fact that the Imaginary-Backward-wavefunction will be immersed into an Expanding Universe, and by default will end up being the real wavefunciton for the period in question namely, 300,000 years after the big-bang function.

A bigger problem for most 'worldline' functions is that you cannot reverse the actual Universe into a compacting/contracting backwards in time arena, if you could then the current Imaginary SH-WF would be relative to the fact that the Universe was in a Contracting phase,and what we are trying to understand now(the-backward-in-time-function)would actually be a new 'FORWARD-TIME-WAVEFUNCTION' you see the two are comprehensively linked by the Universe Phase SH-WF (EXPANDING)and the Universe phase(EXPANDING).

Some say that the SH-WF propergates along the present space-time Expansion, and cannot be made to propergate any other way, placing the SH-WF into any other domain, 'COllAPSES' the very wave-function and only yield distortion or paradoxes.

Any SH-WF imaginary or Real would always end up real, relative to the phase of the Universe it is propergating in, thus a contracting Universe cannot have an Expanding Shroedinger Wavefuncion.

 

[jcsd]

The modulus of a complex number z (|z|), is defined as follows:

$$z = x + yi$$

$$|z| = \sqrt{x^2 + y^2}$$

(Referring to my first post) you can see that both the real and imaginary part of z contribute to the real physical property of the probability of finding a particle at a certain point.

The wave function in quantum physics was derived (though it's imporant to not that this was down by an inductive rather than deductive process, as quantum physics cannot be derived from classical physics) from the classical wave equation for a plane wave. In this classical equation only the real part (x) of the complex quantity (z) is given physical significance.

Advanced waves in which basically solutions to the Schroedinger equation (actually this incorrect as the Schroedinger equation doesn't posess such solutions, but these solutions can be taken from considering the limiting case in the equations of relativistic quantum mechanics)which propagate backwards in time are considered. These backwards in time solutions are the complex conjuagate of the Schoredinger equation, the complex conjugate of a complex number z ($$\inline{\bar{z}}$$) is defined as follows:

$$z = x + yi$$

$$\bar{z} = x - yi$$

 

[jcsd]

I've just realized that I didn't define the symbols:

$$i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V\Psi$$

$$\inline{i}$$ is an imaginary number and is $$\inline{\sqrt{-1}}$$

$$\inline{\hbar}$$ is the ratationlized Planck constant and it has a value of 1.054 x 10^-34 Js

$$\inline{\Psi}$$ is the wavefunction of the particle

$$\inline{t}$$ is time

$$\inline{x}$$ is the postion of the particle

$$\inline{m}$$ is the mass of the particle

$$\inline{V}$$ is the potential

I don't know how familair you are with calculus and partial differentiatin, but $$\inline{\frac{\partial\Psi}{\partial t}}$$ is the partial differential of the wavefunction with respect to time and $$\inline{\frac{\partial^2\Psi}{\partial x^2}}$$ is the second order partial diferential of the wavefunction with repsect to postion.

A complex number is basically a real number and an imaginary number added together and can always be represnted as follows:

$$z = x + yi$$

where x and y are real numbers and $$\inline{i}$$ is equal to $$\inline{\sqrt{-1}}$$

If we ignore the potential (as we would for a free particle)we can represent the wavefunction as:

$$i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2}$$

Therefore the complex conjugate of the wavefunction (see my last post) is:

$$-i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2}$$