Quantum Mechanics: Deriving 3D Schrodinger Wave Equation

Obtaining Three Dimensional Schrodinger Wave Equations

• The first step to obtain three-dimensional three independent Schrödinger wave equations from the time-dependent Schrödinger's equation is to separate the variables by assuming a product solution of the form ψ(x, y, z, t) = X(x)Y(y)Z(z)T(t).

Significance of Three Independent Schrödinger Wave Equations

• Obtaining three independent Schrödinger wave equations is significant because it allows for the description of the behavior of particles in three-dimensional space, enabling the solution of problems in quantum mechanics.

Importance of Spherical Coordinates

• It is important to express the wave function in terms of spherical coordinates when deriving three-dimensional Schrödinger wave equations because it simplifies the mathematical treatment and allows for the exploitation of the symmetry of the problem.

Time-Dependent Schrödinger's Equation

• The time-dependent Schrödinger's equation is a partial differential equation that describes the time-evolution of a quantum system, given by iℏ(∂ψ/∂t) = Hψ, where H is the Hamiltonian operator.
• The time-dependent Schrödinger's equation relates to the three-dimensional three independent Schrödinger wave equations by being the starting point for their derivation.

Key Steps in Deriving Three Dimensional Schrödinger Wave Equations

• Assume a product solution of the form ψ(x, y, z, t) = X(x)Y(y)Z(z)T(t) to separate the variables.
• Substitute the product solution into the time-dependent Schrödinger's equation.
• Separate the variables by grouping the terms corresponding to each coordinate.
• Obtain three ordinary differential equations, one for each coordinate, which are the three-dimensional three independent Schrödinger wave equations.

Significance of Deriving Three Dimensional Schrödinger Wave Equations

• Deriving the three-dimensional three independent Schrödinger wave equations is significant in quantum mechanics because it allows for the solution of problems in three-dimensional space, enabling the calculation of energy eigenvalues and wave functions for particles in various potentials.