The major problem in quantum physics is to find the wave function of a given physical system and to understand (predict) how the wave function of the system evolves in time, or how it changes under external influences. In 1926, Erwin Schrodinger predicted that the wave function of a given physical system might be completely determined if the total energy of the system was known. Since the equation represents a real physical system, it must satisfy the following conditions: the equation must be linear and coefficients appearing in this equation should only depend on the parameters characteristic of the particle. This chapter focuses on to the non-relativistic case only. It considers the case of a free particle moving in one dimension. The chapter shows that the Schrodinger equation can be obtained from the energy of the free particle. It shows that the Schrodinger equation guarantees the conservation of normalization of the wave function.
We study the problem of the relativistic motion of a 1/2-spin particle in an exactly solvable potential, which consists of the harmonic oscillator potential plus a novel angle-dependent potential, The analytic bound state solutions of the Dirac equation for this potential are obtained by using the Nikiforov–Uvarov method. The wave functions of the radial and angle-dependent parts of the Dirac equation are derived in the form of the Laguerre and Jacobi polynomials. The contribution of the angle-dependent potential to the relativistic energy spectra is discussed under the condition that the scalar potential is equal to or minus the vector potential.
The Madelung equations express the Schrödinger equation as a continuity equation and modified Hamilton–Jacobi equation. These equations are equivalent to the Euler equations for a compressible, potential flow, when classical pressure per unit density is replaced by the quantum potential per unit mass. We extend this hydrodynamic interpretation by quantising a single, spinless, non-relativistic particle constrained to a surface wave with small slope. The wave is distinct from the wave function and, in order to reproduce the Schrödinger equation, it must satisfy the kinematic boundary condition for a free surface advected by twice the Madelung velocity field.
We solve the stationary Schrödinger equation for a particle confined to a 3D spherical wedge -- the region
{(r,θ,φ):0≤r≤R,0≤θ≤π,0≤φ≤Φ} with Dirichlet BCs on all surfaces. This exactly solvable constrained-domain model exhibits spectral reorganisation under symmetry-breaking BCs and provides an operator-domain viewpoint on angular momentum quantisation. We obtain three main results. First, the stationary states are standing waves in the azimuthal coordinate and consequently are \emph{not} eigenstates of
L^z; we prove
⟨Lz⟩=0 with
ΔLz=ℏnφπ/Φ=0, demonstrating that angular momentum projection becomes an observable with genuine quantum uncertainty rather than a good quantum number. Second, the effective azimuthal quantum number
μ=nφπ/Φ is generically non-integer, and square-integrability of the polar wavefunctions at both poles requires the angular eigenvalue parameter
ν to satisfy
ν−μ∈Z≥0. This regularity constraint yields a hierarchy: sectoral solutions (
ν=μ, satisfying the first-order highest-weight condition) exist for any real
μ>0, while tesseral and zonal solutions require integer steps, appearing only when
μ itself is integer. Third, application to a Coulomb potential shows that the familiar integer angular momentum spectrum of hydrogen arises from the periodic identification
φ∼φ+2π that defines the full-sphere Hilbert space domain; modified boundary conditions yield a reorganised spectrum with non-integer effective angular momentum. The model clarifies the distinct roles of single-valuedness (selecting integer
m via azimuthal topology) and polar regularity (selecting integer
ℓ≥∣m∣ via analytic constraints) in the standard quantisation of orbital angular momentum.
We calculate exact solutions of the Schrödinger equation for a particle constrained to move along a spherical surface and interacting with non-central potentials, namely, (i) Makarov, (ii) ring-shaped pseudo-harmonic oscillatory, and (iii) Kratzer potentials. We also study exact solutions of the Pauli equation in the same geometrical setting for a charged particle in the presence of a uniform magnetic field. In this case, the two-component spinor can adhere to the surface only if the magnetic field intensity has certain special values. The solutions of Schrödinger equations allow us to obtain exact Pauli spinors and their corresponding energy eigenvalues for the same non-central potentials.
A simple procedure has been found for the general solution of the time-independent Schrödinger Equation (SE) with the help of quantization of potential area in one dimension without making any approximation.Energy values are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and anti symmetric in bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, they should then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and the spherical symmetric central potential well, the relativistic theory of Dirac as examples.