Schrodinger’s Equation for Three Dimensions

QM in Three Dimensions The one dimensional case was good for illustrating basic features such as quantization of energy.

QM in Three Dimensions The one dimensional case was good for illustrating basic features such as quantization of energy. However 3-dimensions is needed for application to atomic physics, nuclear physics and other areas.

Schrödinger's Equa 3Dimensions For 3-dimensions Schrödinger's equation becomes,

Schrödinger's Equa 3Dimensions For 3-dimensions Schrödinger's equation becomes, Where the Laplacian is

Schrödinger's Equa 3Dimensions For 3-dimensions Schrödinger's equation becomes, Where the Laplacian is and Schrödinger's equation in 3D is made up of the 1D equations for the independent axes. The equations have the same form!

Schrödinger's Equa 3Dimensions The stationary states are solutions to Schrödinger's equation in separable form,

Schrödinger's Equa 3Dimensions The stationary states are solutions to Schrödinger's equation in separable form, The TISE for a particle whose energy is sharp at is,

Particle in a 3 Dimensional Box

Particle in a 3 Dimensional Box The simplest case is a particle confined to a cube of edge length L.

Particle in a 3 Dimensional Box

Particle in a 3 Dimensional Box The simplest case is a particle confined to a cube of edge length L. The potential energy function is for That is, the particle is free within the box.

Particle in a 3 Dimensional Box The simplest case is a particle confined to a cube of edge length L. The potential energy function is for That is, the particle is free within the box. otherwise.

Particle in a 3 Dimensional Box Note: If we consider one coordinate the solution will be the same as the 1-D box.

Particle in a 3 Dimensional Box Note: If we consider one coordinate the solution will be the same as the 1-D box. The spatial waveform is separable (ie. can be written in product form):

Particle in a 3 Dimensional Box Note: If we consider one coordinate the solution will be the same as the 1-D box. The spatial waveform is separable (ie. can be written in product form): Substituting into the TISE and dividing by we get,

Particle in a 3 Dimensional Box The independent variables are isolated. Each of the terms reduces to a constant:

Particle in a 3 Dimensional Box Clearly

Particle in a 3 Dimensional Box Clearly The solution to equations 1,2, 3 are of the form where

Particle in a 3 Dimensional Box Clearly The solution to equations 1,2, 3 are of the form where Applying boundary conditions we find,

Particle in a 3 Dimensional Box Clearly The solution to equations 1,2, 3 are of the form where Applying boundary conditions we find, where

Particle in a 3 Dimensional Box Clearly The solution to equations 1,2, 3 are of the form where Applying boundary conditions we find, where Therefore,

Particle in a 3 Dimensional Box with and so forth.

Particle in a 3 Dimensional Box with and so forth. Using restrictions on the wave numbers and boundary conditions we obtain,

Particle in a 3 Dimensional Box with and so forth. Using restrictions on the wave numbers and boundary conditions we obtain,

Particle in a 3 Dimensional Box with and so forth. Using restrictions on the wave numbers and boundary conditions we obtain, Thus confining a particle to a box acts to quantize its momentum and energy.

Particle in a 3 Dimensional Box Note that three quantum numbers are required to describe the quantum state of the system.

Particle in a 3 Dimensional Box Note that three quantum numbers are required to describe the quantum state of the system. These correspond to the three independent degrees of freedom for a particle.

Particle in a 3 Dimensional Box Note that three quantum numbers are required to describe the quantum state of the system. These correspond to the three independent degrees of freedom for a particle. The quantum numbers specify values taken by the sharp observables.

Particle in a 3 Dimensional Box The total energy will be quoted in the form

Particle in a 3 Dimensional Box The ground state ( ) has energy

Particle in a 3 Dimensional Box Degeneracy

Particle in a 3 Dimensional Box Degeneracy: quantum levels (different quantum numbers) having the same energy.

Particle in a 3 Dimensional Box Degeneracy: quantum levels (different quantum numbers) having the same energy. Degeneracy is a natural phenomena which occurs because of the same in the system described (cubic box).

Particle in a 3 Dimensional Box Degeneracy: quantum levels (different quantum numbers) having the same energy. Degeneracy is a natural phenomena which occurs because of the same in the system described (cubic box). For excited states we have degeneracy.

Particle in a 3 Dimensional Box There are three 1st excited states having the same energy. They correspond to combinations of the quantum numbers whose squares sum to 6.

Particle in a 3 Dimensional Box There are three 1st excited states having the same energy. They correspond to combinations of the quantum numbers whose squares sum to 6. That is

Particle in a 3 Dimensional Box The 1st five energy levels for a cubic box. n2 Degeneracy 12 none 11 3 9 6 4E0 11/3E0 2E0 3E0 E0

Schrödinger's Equa 3Dimensions The formulation in cartesian coordinates is a natural generalization from one to higher dimensions.

Schrödinger's Equa 3Dimensions The formulation in cartesian coordinates is a natural generalization from one to higher dimensions. However it not often best suited to a given problem. Thus it may be necessary to convert to another coordinate system.

Schrödinger's Equa 3Dimensions Consider an electron orbiting a central nucleus.

Schrödinger's Equa 3Dimensions Consider an electron orbiting a central nucleus. An obvious coordinate choice is a spherical system centred at the nucleus.

Schrödinger's Equa 3Dimensions Consider an electron orbiting a central nucleus. An obvious coordinate choice is a spherical system centred at the nucleus. This is an example of a central force.

Schrödinger's Equa 3Dimensions The Laplacian in spherical coordinates is:

Schrödinger's Equa 3Dimensions The Laplacian in spherical coordinates is: Therefore becomes .ie dependent only on the radial component r.

Schrödinger's Equa 3Dimensions The Laplacian in spherical coordinates is: Therefore becomes .ie dependent only on the radial component r. Substituting into the time TISE leads to Schrödinger's equation for a central force.

Schrödinger's Equa 3Dimensions Solutions to equation can be found by separating the variables in the Schrödinger's equation.

Schrödinger's Equa 3Dimensions Solutions to equation can be found by separating the variables in the Schrödinger's equation. The stationary states for the waveform are:

Schrödinger's Equa 3Dimensions After some rearranging we find that,

Schrödinger's Equa 3Dimensions The terms are grouped so that those involving a single variable appear together surrounded by curly brackets.

Interlude

Schrödinger's Equa 3Dimensions The uncertainty principal for angular momentum states that:

Schrödinger's Equa 3Dimensions The uncertainty principal for angular momentum states that: it is impossible to specify simultaneously any two components angular momentum.

Schrödinger's Equa 3Dimensions The uncertainty principal for angular momentum states that: it is impossible to specify simultaneously any two components angular momentum. That is, if one component of L is sharp then the other two are “fuzzy”. Momentum is quantised (or sharp).

Schrödinger's Equa 3Dimensions Further we find that L and one component say Lz are quantised.

Schrödinger's Equa 3Dimensions Further we find that |L| and one component say Lz are quantized. Wave functions for which |L| and Lz are both sharp known as spherical harmonics.

End of of Interlude

Schrödinger's Equa 3Dimensions After some rearranging we found that,

Schrödinger's Equa 3Dimensions Each group must reduce a constant for every choice of the variables.

Schrödinger's Equa 3Dimensions Each group must reduce a constant for every choice of the variables. The three equations are:

Schrödinger's Equa 3Dimensions Each group must reduce a constant for every choice of the variables. The three equations are: R

Schrödinger's Equa 3Dimensions Each group must reduce a constant for every choice of the variables. The three equations are: R

Schrödinger's Equa 3Dimensions Solving each equation and applying the appropriate boundary conditions we find that the angular momentum is quantised as follows:

Schrödinger's Equa 3Dimensions Solving each equation and applying the appropriate boundary conditions we find that the angular momentum is quantised as follows:

Schrödinger's Equa 3Dimensions The orbital quantum number must a positive integer. The magnetic quantum number is limited to absolute values not larger than (for a given value of ).

Schrödinger's Equa 3Dimensions The products are the spherical harmonics .

Schrödinger's Equa 3Dimensions The products are the spherical harmonics . The restrictions on the orbital and magnetic quantum numbers are required to keep the spherical harmonics well-behaved.

Schrödinger's Equa 3Dimensions Introducing the separated variables into the TISE, we get:

Schrödinger's Equa 3Dimensions Introducing the separated variables into the TISE, we get: The radial wave equation.

Schrödinger's Equa 3Dimensions The reduction from the Schrödinger's equation to the radial wave equation holds for any central force. The radial wave equation determines the radial part of the wavefunction and the allowed energies.

Schrodinger’s Equation for Three Dimensions Atomic Hydrogen

Atomic Hydrogen We now study the simple case of the hydrogen atom from the view point of wave mechanics.

Atomic Hydrogen Consider an electron of mass m, orbiting the nucleus. The potential energy is given by, Where Z (atomic number) =1 for the hydrogen atom.

Atomic Hydrogen The hydrogen atom is a central force problem, the stationary states are and radial wave function given by the equation,

Atomic Hydrogen The movement of the nucleus be accounted for by replacing the electron mass by the reduced mass, mass electron, Mass nucleus,

Atomic Hydrogen So that Schrödinger's equation (Cartesian and spherical) and the radial wave equation can be written in terms of the rest mass.

Atomic Hydrogen So that Schrödinger's equation (Cartesian and spherical) and the radial wave equation can be written in terms of the rest mass.

Atomic Hydrogen is the potential energy of the electron. The other terms represent the kinetic energy of the electron.

Atomic Hydrogen is the potential energy of the electron. The other terms represent the kinetic energy of the electron. For an electron where all the kinetic energy is orbital and

Atomic Hydrogen Hence, Therefore in the radial wave equation this represents the orbital contribution to the kinetic energy for a mass m with angular momentum L.

Atomic Hydrogen The derivative terms represent the energy of electron motion toward and away from the nucleus.

Atomic Hydrogen The derivative terms represent the energy of electron motion toward and away from the nucleus. The leftmost term is like the expression for the kinetic energy of a matter wave moving along r.

Atomic Hydrogen However the term is not consistent with motion only along r.

Atomic Hydrogen However the term is not consistent with motion only along r. Instead it is consistent with motion described by .

Atomic Hydrogen However the term is not consistent with motion only along r. Instead it is consistent with motion described by . And we can show that

Atomic Hydrogen Therefore the radial wave equation for a pseudo-wavefunction takes the form of the Schrödinger’s equation 1D,

Atomic Hydrogen Therefore the radial wave equation for a pseudo-wavefunction takes the form of the Schrödinger’s equation 1D, With effective potential energy

Atomic Hydrogen From the Schrödinger’s equation for we get Which is in exact agreement Bohr’s theory.

Atomic Hydrogen From the Schrödinger’s equation for we get Which is in exact agreement Bohr’s theory. From the quantum mechanics we have shown that Bohr’s assumptions were correct.

Atomic Hydrogen n is the principle quantum number is the Bohr radius

Atomic Hydrogen Recall

Atomic Hydrogen Recall Although n can be any integer, Script L is restricted to less than (n-1)

Atomic Hydrogen Recall Although n can be any integer, And (doesn’t change) M script L remains the same

Atomic Hydrogen The radial waves R(r) for hydrogenic atoms are products of exponentials with polynomials in r/a0.

Atomic Hydrogen The radial waves R(r) for hydrogenic atoms are products of exponentials with polynomials in r/a0. The radial wave functions are tabulated as Rn,l(r).

1 2 3

Atomic Hydrogen For historical reasons, states with the same principal quantum number form a shell. Shells are: K,L,M,N,O,…(1,2,3,...) States have both n and l the same are called sub-shells. l=0,1,2,3,… are designated by s,p,d,f,… .

Hydrogenic Atomic The ground state of a one electron atom with atomic number Z ( ),

Hydrogenic Atomic The ground state of a one electron atom with atomic number Z ( ), The wave function for this state is

Hydrogenic Atomic The ground state of a one electron atom with atomic number Z ( ), The wave function for this state is The electron described by this wave is found with probability per volume,

Hydrogenic Atomic The probability distribution is spherically symmetric. That is, it the likelihood of finding the electron is the same for all points equidistant from the nucleus.

Hydrogenic Atomic The probability distribution is spherically symmetric. That is, it the likelihood of finding the electron is the same for all points equidistant from the nucleus. Therefore it is convenient to define the radial probability (and its probability density P(r)).

Hydrogenic Atomic The probability distribution is spherically symmetric. That is, it the likelihood of finding the electron is the same for all points equidistant from the nucleus. Therefore it is convenient to define the radial probability (and its probability density P(r)). The probability of finding the electron in a spherical shell of radius r, thickness dr.

Hydrogenic Atomic For the hydrogenic 1s state,

Hydrogenic Atomic For the hydrogenic 1s state,

Hydrogenic Atomic The probability density is a “cloud” for the electron in 1s state.

Hydrogenic Atomic The normalization becomes, The integral over all possible values of r.

Hydrogenic Atomic The normalization becomes, The integral over all possible values of r. The avg distance of the electron from the nucleus is weight of each possible distance with the probability of it being found there:

Hydrogenic Atomic In general the avg of any function of distance f(r) is given by,

Hydrogenic Atomic – Excited States There are four 1st excited states for hydrogenic atoms: They have the same principle quantum number n=2 and hence the same energy, Therefore the 1st excited state is fourfold degenerate.

Hydrogenic Atomic – Excited States The 2s state is spherically symmetric. However the other three states belong to the 2p sub-shell and are not spherical symmetric.