The Schrödinger–Hemholtz Theorem states that any quantum mechanical wave function, ψ, can be expressed as a linear combination of single-mode wave functions, αnm, known as wavelets:
ψ = ∑n,mαnmαnm
Schrödinger–Hemholtz Theorem is used in various areas of quantum mechanics, most notably in the study of atomic structure and interaction with electromagnetic radiation. The theorem is also used frequently in the field of condensed matter physics, to understand the behavior of electrons in solids.
The Schrödinger–Hemholtz Theorem was first presented in 1932 by Erwin Schrödinger, in a paper entitled “On The Coordinate Dependence of Wave Functions”, and later extended by Wilhelm Hemholtz in the same year. The theorem is important for the development of the mathematical theory of quantum mechanics, and its predictions have been experimentally confirmed in several different areas.
In the paper where the theorem was first proposed, Schrödinger began by defining the wave function, Ψ, for a generic quantum system through the Schrödinger equation:
ℏΨ = μ(Δ2 + V(r))Ψ
where ℏ is Planck’s constant, μ is the appropriate mass, V is the potential energy, and r is the position vector. He then postulated a general solution to this equation, which can be written as:
Ψ = ∑n,mαnmψnm
where ψnm is a single-mode wave function, and αnm is a coefficient that describes the amplitude and phase of the mode.
Once Schrödinger had postulated this general solution, he set out to prove that it is indeed the valid form of the wave function. He proceeded by considering a discrete set of coordinate points, each with a value of Ψ and Ψ′ (the first and second derivatives of Ψ), for any arbitrary quantum system. By applying a numerical integration method, he showed that the coefficients αnm completely describe the Fourier–Bessel expansion of the wave function, Ψ:
Ψ = ∑n,mαnmJnm(kr)
where Jnm is a Bessel function, and k is the wave vector associated with the frequency of the mode. By repeating this step and including the higher-order derivatives of Ψ, he showed that all of the coefficients, αnm, must be present in the Fourier–Bessel expansion of the wavefunction if it is to contain all of the information. As a result, Schrödinger concluded that one can reconstruct the wave function, Ψ, of any system by knowing only the two sets of coefficients, αnm, and their corresponding frequencies. This is the essence of the Schrödinger–Hemholtz Theorem.
The importance of the Schrödinger–Hemholtz Theorem is that it provides a basis for understanding both the behavior of the electrons in atoms and molecules, as well as the electromagnetic spectrum of light. In other words, the theorem allows one to make predictions as to how the wavefunction of a quantum system will evolve when exposed to an external field. The theorem has been used extensively in the study of the response of atoms to light, as well as in the study of collisional processes between atomic and molecular states.
The Schrödinger–Hemholtz Theorem remains an important tool in the arsenal of quantum mechanics. By allowing the behavior of quantum systems to be predicted without the need for extensive numerical calculations, it has enabled physicists to make accurate predictions about the behavior of certain systems. Despite its importance in the field of quantum mechanics, however, the theorem continues to be challenged by some scientists, as the assumptions made by Schrödinger and Hemholtz are still open to dispute. Nevertheless, it remains a cornerstone of quantum mechanics and its importance continues to be recognized today.
