Show the principle steps leading from Schrodinger equation in Cartesian coordina

Question

Show the principle steps leading from Schrodinger equation in Cartesian coordinates to the radial form of the time independent Schrodinger equation. I expect that you show and/or discuss: a) how the explicit form of the di?erential equation for Ri(r) arises; b) what domain take the spherical coordinates and why; c) what boundary conditions are introduced for the wave function that lead to quantization of the spectrum; d) you introduce the quantum numbers you expect to occur and pinpoint their origin; e) describe in detail the normalization of the wave function in spherical coordinates; f) show how one absorbs the radial volume element of the normalization integral into the form of the radial invention; g) show the steps leading to the modi?ed radial Schrodinger for this modi?ed radial wave function. Make an e?ort to be concise yet complete.

Explanation / Answer

Time-dependent equation The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation, which gives a description of a system evolving with time[2] : Time-dependent Schrödinger equation (general) i hbar rac{partial}{partial t}Psi = hat H Psi where ? is the wave function of the quantum system, i is the imaginary unit, h is the reduced Planck constant, and hat{H} is the Hamiltonian operator, which characterizes the total energy of any given wavefunction and takes different forms depending on the situation. A wave function which satisfies the non-relativistic Schrödinger equation with V=0. In other words, this corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here. The most famous example is the non-relativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field): Time-dependent Schrödinger equation (single non-relativistic particle) ihbarrac{partial}{partial t} Psi(mathbf{r},t) = left [ rac{-hbar^2}{2m} abla^2 + V(mathbf{r},t) ight ] Psi(mathbf{r},t) where m is the particle's mass, V is its potential energy, ?2 is the Laplacian, and ? is the wavefunction (more precisely, in this context, it is called the "position-space wavefunction"). In plain language, it means "total energy equals kinetic energy plus potential energy", but the terms take unfamiliar forms for reasons explained below. Given the particular differential operators involved, this is a linear partial differential equation. It is also a diffusion equation. The term "Schrödinger equation" can refer to both the general equation (first box above), or the specific nonrelativistic version (second box above and variations thereof). The general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in various complicated expressions for the Hamiltonian. The specific nonrelativistic version is a simplified approximation to reality, which is quite accurate in many situations, but very inaccurate in others (see relativistic quantum mechanics). To apply the Schrödinger equation, the Hamiltonian operator is set up for the system, accounting for the kinetic and potential energy of the particles constituting the system, then inserted into the Schrödinger equation. The resulting partial differential equation is solved for the wavefunction, which contains information about the system. Time-independent equation Each of these three rows is a wavefunction which satisfies the time-dependent Schrödinger equation for a harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wavefunction. Right: The probability distribution of finding the particle with this wavefunction at a given position. The top two rows are examples of stationary states, which correspond to standing waves. The bottom row an example of a state which is not a stationary state. The right column illustrates why stationary states are called "stationary". The time-dependent Schrödinger equation predicts that wavefunctions can form standing waves, called stationary states (also called "orbitals", as in atomic orbitals or molecular orbitals). These states are important in their own right, and moreover if the stationary states are classified and understood, then it becomes easier to solve the time-dependent Schrödinger equation for any state. The time-independent Schrödinger equation is the equation describing stationary states. (It is only used when the Hamiltonian itself is not dependent on time.) Time-independent Schrödinger equation (general) EPsi=hat H Psi In words, the equation states: When the Hamiltonian operator acts on the wavefunction ?, the result might be proportional to the same wavefunction ?. If it is, then ? is a stationary state, and the proportionality constant, E, is the energy of the state ?. The time-independent Schrödinger equation is discussed further below. In linear algebra terminology, this equation is an eigenvalue equation. As before, the most famous manifestation is the non-relativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field): Time-independent Schrödinger equation (single non-relativistic particle) E Psi(mathbf{r}) = rac{-hbar^2}{2m} abla^2 Psi(mathbf{r}) + V(mathbf{r}) Psi(mathbf{r}) with definitions as above.

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