In quantum mechanics: Wave function does not have any physical meaning. Mathematically wave function must define a complete set of functions (orthonormal set of functions), but the square of the absolute of the wave function is proportional to the probability of finding the particle (in quantum its equivalent to the wave function) in the state represented by a point in that space. It is a solution of Schrodinger equation. The detail of this may be found in any text in quantum mechanic like Quantum mechanics by Powel and Craseman

In Wave motion (either elastic wave or electromagnetic wave) the wave function represent the physical quantity that constitute the propagating wave (in EM wave it represent the Electrical field intensity perpendicular to the magnetic field intensity variation with time and in elastic wave it represent the variation of the displacement of atoms of the medium that the wave travels, from its equilibrium position with time. It is solution of the wave function. The detail of this topic can be found in any text in optics and wave motion.

Hi Sridhar,

The wave function is the space-time representation of (the knowledge of) the state of a system having some wave-like behavior. This definition is equally valid in classical mechanics as well as in QM.

For understanding, go through Dirac's monograph (The principles of QM) and Feynman's lect. (vol. III) and for history go through any other good old book on QM. as suggested above by Francis and Faiq.

Thank you very much sir,

1) In matter waves, the particle displacement is described as

Ψ=A e^{i (kx - w t )}

2) but in waves in stretched string, displacement is described as

Y=A cos(kx - w t )

we have neglected imaginary part in stretched string. why should not we neglect the imaginary part in matter waves?. I am really confused in imaginary part.

Sridhar

As i have mentioned above , in elastic wave (Y=A cos(kx - w t )) , Y represent the displacement of atomic sites from its equilibrium position (microscopic point of view) or sometimes pressure wave (macroscopic) you know that the displacement is a real physical quantity, while in Q. M. no matter about the wave function itself, what is important is the square absolute of the wave function which is proportional to the probability, you know that probability is a real quantity hence we use absolute square of the wave function IΨI ^2

regards

Wave function, in quantum mechanics, variable quantity that mathematically describes the wave characteristics of a particle. The value of the wave function of a particle at a given point of space and time is related to the likelihood of the particle’s being there at the time. By analogy with waves such as those of sound, a wave function, designated by the Greek letter psi, Ψ, may be thought of as an expression for the amplitude of the particle wave (or de Broglie wave), although for such waves amplitude has no physical significance. The square of the wave function, Ψ•Ψ, however, does have physical significance: the probability of finding the particle described by a specific wave function Ψ at a given point and time is proportional to the value of Ψ•Ψ.

By definition, wave functions are solutions to a family of linear second order partial differential equation called wave equations. We most often associate wave functions with solutions of Schrodinger's equations, which is one such equation. Many other wave equations exist both in and outside of quantum mechanics.

By introducing more variables a wave equation may also take the form of a group of first order PDEs of a set of quantities(notably, this includes Maxwell's and Dirac's equations). But in any case they have to be linear PDEs.

Wave like behavior is a result of the mathematical structure of the problem. The partial derivatives generates oscillatory behavior. Maybe more importantly, because the wave equations are linear, the linear combination of a set of solutions is still a solution--this results in the superposition principle. Such superpositions and oscillatory behaviors are common feature of all kinds of waves: strings, sound, light, electron motion, etc.

There is a long history of research about wave like phenomenon and the mathematics of it is well established, so when new experimental results started to suggest particle-wave duality, people immediately know there probably should be a wave function. Schrodinger started to try out all types of wave equations, and in 1926 published the correct one for electron that described hydrogen atom spectral lines. It is noteworthy that he also tried Klein–Gordon equations, which where correct for spinless particles, but not for electrons. In fact, Schrodinger wasn't the first one to come up with the equation either, but the other guy couldn't get his paper published, and as a result now I cannot remember his name.

A wave function is a function of the coordinates in configuration space and time, describing the state of a physical system. The square of the modulus of the wave function gives the density of probability of finding the configuration of the system in a neighbourhood of a point in configuration space. It's supposed to be the solution of a partial differential equation: i h \partial_t \Psi = \hat H \Psi, where \Psi is the Hamiltonian operator. This statement is a bit strange, because the wave function is not supposed to have a direct physical interpretation and, as a consequence, the statement that " i h \partial_t \Psi = \hat H \Psi " is not true in the same sense that it is true that "Sridhar Sampath asked: Can someone tell me the exact definition of Wave function?" The last statement can be verified immediately, at least for those in researchgate. The statement " i h \partial_t \Psi = \hat H \Psi " is verified, indirectly, because some of its logical consequences are confirmed, on the assumption that the axioms of quantum mechanics are true.

Dear Sridhar,

Adding to what already has been said I also recommend you to look at my comments and discussions with Demetris Christopoulos in connection with his question: What is your opinion about the Sturm-Liouville theory?

The standard definition is straightforward either employing the appropriate differential equations and its associated boundary conditions or using the topological definition of properly defined linear spaces (Hilbert Spaces).

The physical meaning of the wavefunction is a different business, although Born's statistical interpretation of the absolute square being related to probability densities is not in itself controversial. However the physical maning of the wavefunction is a more controversial subject. Can it be seen or directly measured or is it the associated charge density which is of observational support? For interesting discussions on this problem, see Eric Scerri’s website, in particular his educational paper ”Have Orbitals Really Been Observed?”.

Hi Sridhar,

It seems most people are leaving aside your second question posted in reply to my reply. Probably they think it is meant for me to reply. Well, let me reply then.

As you know the well-known differential equations describing wave motion are linear, that is any linear combination of two solutions will be a valid solution of the wave equation.

Thus, for the classical string case, the sine and cosine are both valid solutions differing in phase by \pi/2. They can very well be superposed as (cos + i sin ) to have more general solution, with the understanding that only the real part OR the imaginary part can have any physical significance, not both simultaneously. You can combine these two superpositions to get back the original sine and cosine forms.

Next when we come to QM, the same logic goes through, but in stead of Re(\psi) OR Im(\psi) we have the full wave function \psi as the valid solution. How to extract physical significance then? Born proposed that (\psi* \psi ) be interpreted as the probability of finding the particle. This was the third way available to have a real quantity from the complex \psi.

The \psi* is however not a solution of the Schrodinger equation. It is a solution of the conjugate Schrodinger equation. The reason why this is so is that, the time-dependent Schrodinger eqn. is 1st order in time derivative having i(\hbar)(d\psi) /dt = H \psi. Thus, it remained for along time an unresolved issue and many attempts have been made to make sense of "why the Born rule works so perfectly well?"

In my opinion a better understanding is provided by Cramer's transactional interpretation which interprets \psi* as a backward-in-time solution which helps resolve many paradoxes and contradictions in QM.

I have taken it one step further in my Psychophysical Interpretation of quantum theory to establish the role of the conscious observer. You can download and see the paper from arXiv.org. It addresses the question raised by you namely, the imaginary part or equivalently the \psi*.

It is a pity that very few people really bother about these tricky issues in the interpretation of QM and are happy with applying QM to get solutions.

The question that should have been asked of Born was that what was the interpretation of \psi* and it being a solution of the conjugate Schrodinger equation, what business it had with a solution of the Schrodinger equation. These are two distinct equations having opposite temporal behavior. Just because \psi* \psi is real it need not correspond to reality. But the fact is it does correspond to reality. And you see right up to Cramer, no one bothered to interpret \psi*. Everyone was happy interpreting \psi as giving the prob. amp. for the particle to be at (x,t), and \psi* as simply the mathematical complex conjugate. There is in fact much more to it, and the interpretation problem of QM is not over yet !!

Regards to one and all..

The short answer to your question is that a wave function r e p r e s e n t s the state of physical system, in the position representation. In the latter representation the position operator, which is an observable, is "diagonal," (more precisely, coincides with the multiplication operator by identical function.) The wave function is not observable, for instance, since complex-valued and since there are other representations, like the momentum representation ["Observables" in quantum are linear self-adjoint o p e r a t o r s in Hilbert Spaces, and what is really observed are spectral measures corresponding to such operators and elements in a state space.

The Hilbert space is a "dummy" space. ]

To keep things short, one general advice. Quite often the ideas physicists have in mind about nature are incorrect or at least contradictory, but once it comes down to c a l c u l a t i o n s , results are usually correct. The reason is that the mathematical backup of quantum theory is absolutely firm. On the other hand, that background is relatively subtle and (obviously) not known by the majority of physicists. For a f u l l understanding of quantum theory a basic training in physics + a substantial background in operator theory/functional analysis is necessary.

Best regards, Horst Beyer.

Dear Horst Beyer,

could you suggest me any good books to understand operator theory/functional analysis

Dear Sridhar,

The first books that come to my mind are

"M. Reed and B. Simon: Methods of Modern Mathematical Physics," Vol. I + II.

Vol. I provides the basic functional analysis background of quantum theory. Vol. II provides the background in self-adjoint operators. To my opinion, they are still the best for this purpose. [Vol. III is on mathematical scattering theory and Vol. IV is on spectral theory. ]

Best regards, Horst Beyer

Is it possible to write the wave function in 10 dimension and to solve Schrodinger equation for 10 dimension ?

A good starting book for a discussion of the ontological issues associated with wave functions is the collection of essays collated by Alyssa Ney 'The Wave Function'

If you're more interested in the associated math any good book on QM should help (the 2 volume Cohen-Tannoudji book is pretty good, but there are quite a few good others)

You will find the history of the introduction of the wave function here and its possible relation to a 10 dimensions space and vectorial geometry here:

http://file.scirp.org/pdf/JMP_2018042716061246.pdf

Sridhar Sampath, i understand that you are confused about the imaginary part of the wave equation. this link should help you. http://physics.mq.edu.au/~jcresser/Phys201/LectureNotes/SchrodingerEqn.pdf

read between equation 6.7 and 6.8 which specifically addresses the stretched string vs the harmonic wave function for a free particle of energy E and momentum p, It is basically because the wave function eqn.( 6.8 in the link) is a solution to the schrodinger equation.

Dear Sridhar,

The wave function was initially introduced by Schrödinger to represent an axial resonance state into which de Broglie had concluded that the electron was stabilized into when in the ground state of the hydrogen atom.

You will find the detail of how this historical event proceeded here, with references to the discoverer's own accounts:

http://file.scirp.org/Html/17-7503469_84158.htm

Best Regards

André