Plane waves are fundamental concepts in quantum mechanics and are often used to describe free particles. However, there's a common misunderstanding that plane waves themselves are wavefunctions, which fulfill all the required criteria for a physical state in quantum mechanics. This confusion can be attributed to several factors, which this discussion aims to explore:
Your insights and experiences on this topic will contribute to a broader understanding of the details and common misconceptions in quantum mechanics. Let's explore this subject and clarify why plane waves, despite their importance, are not considered true wavefunctions.
If you set up the Schrödinger equation for a free particle, so the potential energy is V=0, a plane wave indeed solves this equation mathematically.
As you already indicated, this function does not fulfill all criteria for a physically valid wavefunction, particularly because it can't be normalized. This fact is to some extent pointed out by most decent textooks I have come across, but there are also quite a few which fail to mention that properly.
In reality, there is of course no particle that's totally free of any potential.
Having said that, there is a reason why they are still many derivations which use plane waves: in scattering experiments, the initial and final states may behave rather close to what a plane wave would be and therefore using plane waves to approximate these states is the only feasible way to reduce the scattering problem to an analytically soluble equation.
For more on this, see
Article Plane waves and wave packets in elementary quantum mechanics problems
A plane wave can be a wave function.
For ex. As an eigenfuction of the
Kinetic energy part alone.
H=p^2/2m
p=-i hbar d/dx
Psi= exp(ikx)
So
H psi=(hbar k)^2/2m psi
hbar = h/(2 pi)
Plane waves are not true wavefunctions because they do not meet the normalizability criterion essential for physical interpretation in quantum mechanics. Their role is fundamental as mathematical tools and idealized momentum eigenstates, but their direct association with wavefunctions arises from simplifications in teaching and the abstract nature of the mathematics involved.
What's a wave function by definition?
A wavefunction in quantum mechanics must meet specific criteria:
Normalization: The integral of over all space must be finite and equal to 1, ensuring a valid probability density interpretation.
Physical Meaning: It represents the probability amplitude, where gives the probability density for finding a particle in a specific region.
Boundary Conditions: It must satisfy the boundary conditions for the system, e.g., decay at infinity for bound states.
planewave on the other hand...
It is not normalizable over infinite space since is constant and does not decay.
Thus, while it satisfies the Schrödinger equation, it does not represent a physical quantum state.
Quantum mechanics has two essential components:
While operators can be Hermitian or unitary, states are categorized as eigenstates or non-eigenstates. Quantum states are closely tied to symmetry and group theory, forming irreducible representations of specific symmetry groups.
A plane wave is the eigenstate of the linear momentum operator, analogous to how spherical harmonics are eigenfunctions of the angular momentum operator. Plane waves represent the irreducible representations of the translation symmetry group, which describes how a system behaves under spatial translations. According to Noether's theorem, the translation symmetry is associated with momentum conservation. Since the translation symmetry group is Abelian, it has one-dimensional irreducible representations, and these are the plane waves.
Plane waves are widely used in quantum mechanics because they are eigenfunctions of momentum and exhibit the property that acting on them with the momentum operator gives an eigenvalue times the same function. However, plane waves are not physical wavefunctions because they are not normalizable over infinite space. Instead, they serve as idealized mathematical tools for understanding symmetries and building physical states, such as wave packets, which are superpositions of plane waves.
The confusion about plane waves arises because they are often introduced in textbooks and lectures as solutions to the Schrödinger equation for free particles. This simplification emphasizes their utility in representing momentum eigenstates but may lead to the misunderstanding that they are physical wavefunctions. While their role in quantum mechanics is crucial, it is essential to distinguish between their mathematical significance and the criteria that physical wavefunctions must fulfill.
Hi Ahmad, my response is oriented to the interpretation issue. As all know, Born correctly interpreted Schrödinger's wavefunction. The mathematical value of Psi^2 is a probability of presence. This says a lot about what Psi itself is. It is related to a presence in 3D and since 2021 there is a Theory of space that says that physical existence oscillates between 3D and the 4th dimension. All the other theories consider at least these four dimensions, some even up to 11 dimensions, but none of them considers the possibility of an oscillation between these dimensions. They always consider all of them together at the same instance. WHAT IF this oscillation is described by the wavefunction? That will give this math a physical meaning as it matches perfectly with the proven probability observed in 3D. It is an issue of an oscillating presence or existence at 3D. The interference is understood as destructive or "not probable to be" when a part of the quantum system is at 3D in contact with its other part at the 4th dimension. It will also explain how locality is conserved by entangled particles; they are local at the 4th D even when its 3D presence is way apart. I have attached a recent paper that has some references to previous papers. Hope they will inspire you in your research. Regards
This "paper" clearly is misusing the terms of physical dimensions.
It would never have been accepted by a publisher with serious peer review, this IJSR has a fake impact factor ticking one of the biggest boxes for a predatory journal.
Iftikhar Ahmad
Plane waves are often misunderstood as wavefunctions in quantum mechanics because they are fundamental solutions to the Schrödinger equation for free particles and are widely used in theoretical contexts. However, while plane waves, described by ψ(x)=ei(kx−ωt)\psi(x) = e^{i(kx - \omega t)}, satisfy the Schrödinger equation, they do not meet the essential criteria for a physical wavefunction. A true wavefunction must be normalizable, meaning its integral over all space, ∫∣ψ(x)∣2dx\int |\psi(x)|^2 dx, must converge to one, ensuring a well-defined probability interpretation. Plane waves, extending infinitely in space with a constant magnitude, fail this criterion because their probability density is uniform and cannot be normalized. This distinction is crucial but is often blurred in introductory quantum mechanics courses.
One reason for the confusion is the way plane waves are presented in textbooks and lectures. They are often introduced early in quantum mechanics to describe free particles because of their simplicity and utility in mathematical derivations, such as in Fourier analysis and the construction of wave packets. This pedagogical approach, while effective for teaching basic concepts, can inadvertently give students the impression that plane waves themselves are physical states. Textbooks may emphasize their role as eigenfunctions of the momentum operator without sufficiently clarifying that they are idealized constructs rather than physically realizable wavefunctions.
Mathematically, plane waves serve as a basis for representing more general wavefunctions, such as localized wave packets. Any physical wavefunction can be expressed as a superposition of plane waves through Fourier transforms. However, the infinite extent and uniform probability density of plane waves are non-physical, and their true role is as idealized components of broader solutions. This mathematical utility often leads to the misunderstanding that they are standalone physical states, especially when their practical limitations are not emphasized.
The physical interpretation of plane waves also differs significantly from that of true wavefunctions. A plane wave represents a particle with a precisely defined momentum and energy, but its position is completely undetermined, as the probability density is spread uniformly across all space. This contrasts with a localized wave packet, where both momentum and position are described within the limits of the Heisenberg uncertainty principle. Thus, while plane waves are useful for describing momentum eigenstates, they do not correspond to physical states that could be observed in an experiment.
Common analogies, such as describing plane waves as "ripples" or "pure tones," can further perpetuate the misunderstanding. While these analogies are helpful for conceptualizing the momentum aspect of quantum particles, they often downplay the necessity of normalizability and the probabilistic interpretation of wavefunctions. Ultimately, clarifying the distinction between plane waves and true wavefunctions requires emphasizing the idealized nature of plane waves and their role in constructing physical solutions, ensuring that students and practitioners grasp their proper place in quantum mechanics.
The “wave/particle” duality, “plane wave”, “relativistic mass” etc. are opportunistic notions of causality-based official physics to deny the “Evil Quanta”, which made its appearance by the turn of the 20th century. This was a deliberate, subjective and an opportunistic effort (thanks to Kant)– to “deny knowledge in order to make room for faith”
"Nothing exists except atoms and empty space. Everything else is opinion” so said the early Greek dialectical materialist Democritus (~ 420 B.C.). The development of dialectics by the time of Hegel and its extension to the quantum phenomena, now ascertain that the “empty space” of Democritus is not empty, but is full of momentarily existing “VIRTUAL” particle/antiparticle pairs eternally flipping in and out of existence; but serves as the source and the sink of “REAL” quantum particles; mediated by dialectical chance and necessity; in this Infinite, Eternal and Ever-changing universe.
The inconsistency of plane wave and the wave function and the ensuing confusion and jugglery of mathematical idealism in modern theoretical physics and cosmology, arises from the promotion of thought (mind, mathematics, a core premise of idealism and theology) being more fundamental over the physical reality of (particulate) matter in motion; as dialectics from the early Greeks insisted. In physics it started, with Isaac Newton’s the very first sentence of the Foreword of his Principia: “Now that (since Bacon) the substantial forms (of the Aristotelians) have been abandoned from natural philosophy, mathematics should replace them to the maximum possible extent”. It is evident that Newton clearly rejected materialist and conceptual (Begrieff) methods of philosophical enquiry of Aristotle in favour of the mathematical idealism of Plato; which posits that mathematical forms define the phenomenology of objective reality or at least the phenomena described by the sciences have a mathematical structure. In philosophy, I. Kant promoted subjective idealism of thought-derived logical/mathematical categories over physical reality, which for Kant is an 'unknowable thing-in-itself”. Kant justified his subjective idealism in an straight forward way, “I had to deny knowledge in order to make room for faith.” ― Immanuel Kant, Critique of Pure Reason
Following Kant’s subjective idealism and particularly after the recognition of the causality busting and spooky “Evil Quanta”; Albert Einstein raised Newton’s mathematical idealism to its epitome, with the following words, “Our experience hitherto justifies us in believing that nature is the realization of the simplest conceivable mathematical ideas. I am convinced that we can discover by means of purely mathematical constructions the concepts and the laws connecting them with each other, which furnish the key to the understanding of natural phenomena. … In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed” A. Einstein, “Essays in Science”, Translated by Alan Harris from “Mein Weltbild, Quedro Verlag, Amsterdam, 1933), The Wisdom Library, N.Y., p48 – 49, (1934).
Physical (objective) reality had to be redefined to conform to subjective idealism, by abandoning classical materialism of Aristotle. For Einstein, “Since the theory of general relativity (GR) implies the representation of physical reality by a continuous field, the concept of particles and material points cannot play a fundamental part and neither can the concept of motion. The particle can only appear as a limited region in space in which the field strength or energy density is particularly high”. Einstein, A. On the General Theory of Relativity, in David Levy (Ed.). The Scientific American Book of the Cosmos, N.Y., 2000, pp. 13.
So “continuous field” as opposed to (particulate) matter became the Warcry against the “Evil Quanta” in theoretical physics and cosmology – hence “Matter is a Myth”. Since wave is the only form of motion in a continuous (fluid) field and since Maxwell’s electrodynamics based on EM-wave is a satisfactory description of this phenomena, quantum fluid dynamics must be a representation of continuous physical reality and the only way to subjugate the discrete “quanta”. Matter particle in this view can only be a temporary wavelet or drops of such fluids (energy density) creating the “plane wave”.
For official theoretical physics, the continuous field (Minkowoski “spacetime”, EM fields, quantum field) etc., adequately described by geometric, symmetric, mathematical consideration and the “consistency” in their use can be the only way to describe and to know the world – “Our Mathematical Universe” as Max Tegmark the MIT physicist, now claims! But it a subjective universe (far removed from REAL physical one we live in) of the abstract thought realm of certain, deterministic, causality abiding world of Plato. The spooky quanta have to be subjugated within the sticky quantum field; if necessary using arbitrary mathematical tricks like renormalization, limits (finite) etc. Only this way as Kant said one can “deny knowledge (the reality of the ‘Evil Quanta’) in order to make room for faith”.
Faith is an important aspect of managing a class society; by a dominant ruling class. So after the theological faith was disrupted by the Copernican revolution, the new virtual world of mathematics created from Newton to Einstein had to reform or replace the discredited old faith; under the capitalist ruling order.. “Quō Vādis Theoretical Physics and Cosmology? From Newton's Metaphysics to Einstein's Theology”! https://www.peertechzpublications.us/amp/article/view/AMP-6-181/pdf
New Physics – The Negation of Einstein’s Theories of Relativity - The Real Phenomenology of Space-Time-Matter-Motion. JOURNAL OF ADVANCES IN PHYSICS, 22, 54–61. https://doi.org/10.24297/jap.v22i.9594
“KEPLER – NEWTON – LEIBNIZ – HEGEL : Portentous And Conflicting Legacies In Theoretical Physics, Cosmology And In Ruling “: https://rajpub.com/index.php/jap/article/view/9106
The Infinite: http://www.e-journal.org.uk/shape/papers/Special%2064.pdf
Most of the modern-day cosmological myths were fabricated at the turn of the 20th century; mainly through the brain-cooked (no relevance to objective reality) mathematical tools of Lorentz Transforms; to combat the "Evil Quanta" and at the behest of monopoly capitalism and theology!
“The Mystery of the Lorentz Transform: A Reconstruction and Its Implications for Einstein's Theories of Relativity and cosmology” : INSPIRE>HEP: https://inspirehep.net/literature/2158754
What is Light, Really? A Quantum Dialectical View. Ann Math Phys. 2024;7(3):292-299. https://www.mathematicsgroup.us/articles/AMP-7-235.pdf
https://www.researchgate.net/publication/385343389_What_is_Light_Really_A_Quantum_Dialectical_View
Please Note: Some of the links in the Internet of the publications by this author may be blocked, deleted and/or infected with malware; as hostile acts against this author by some internationally coordinated powerful elements of official science. These are attempts to censor, disrupt and/or silence the propagation of effective dialectical views in modern theoretical physics and cosmology!
“It would never have been accepted by a publisher with serious peer review, this IJSR has a fake impact factor ticking one of the biggest boxes for a predatory journal.”
This is an ecclesiastical statement that allows only Kosher “knowledge” to be published and denying freedom of expression of critical ideas!
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