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Spherical Polar Coordinate System

The point P(x,y,z) in spherical coordinate system are expressed in terms of
Distance OP=r (radial coordinate)
θ=Zenith angle made by OP with z-axis.
ϕ=Azimuthal angle made by projection OM of OP on xy-plane with x-axis.
Three coordinates are expressed as:

P(x,y,z)

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Laplacian in Spherical Polar Coordinate System

The Laplacian in cartesian and spherical polar coordinate system is expressed as:

Schrodinger’s equation for spherically symmetric potential is:

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Splitting of Schrodinger’s Equation

Let’s consider that the wavefunction is product of three explicit functions given as:

Putting eq. (2.180) in eq. (2.179), we get:

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The RHS of the eq. (2.181) is dependent purely on ϕ while LHS has dependence on r and θ. Hence this equation holds true for all values of r, θ and ϕ if and only if both sides are equal to some constant (say k²). Hence we can write:

The eq. (2.182) is purely dependent on ϕ and hence is called Φ(ϕ)-equation.

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The eq. (2.183) has dependence on r and θ. The eq. (2.183) can be rewritten by dividing whole equation by sin²θ as:

The eq. (2.186) has LHS purely dependent on r while RHS is purely dependent on θ. Hence each side must be equal to a common constant which in this case is taken to be l(l+1). We can write two equations as:

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3-Components of Schrodinger’s Equation

Three component equations are:

It can be observed that Θ(θ) and Φ(ϕ) equations do not have energy or potential term which implies that these equations yield same information for all kinds of spherically symmetric potentials.

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Quantized States & Energies �in Hydrogen Atom

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Solution of Φ(ϕ) Equation

The solution of equation (2.182) is expressed, with k=m as:

Using Normalization condition, we get:

The component Φ(ϕ) of the wavefunction is well behaved as it is
Single valued
Finite
Continuous everywhere.

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We can observe that φ(0)=φ(2π) which implies that

The m is called the magnetic quantum number which has integral permissible values.
The angle ϕ can take certain values defined by ϕ=2πm and hence is space quantized.

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Solution of Θ(θ) Equation

The Θ(θ) equation (with k=m) is given as:

To change the eq. (2.186) to standard form, we proceed as:

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Putting eq.(2.187) in eq. (2.186), we get:

Equation (2.186) takes the form:

Comparing the eq. (2.188) with standard form of Legendre’s differential equation, we have:

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The solution of the Legendre’s equation is given as:

The solution of Legendre’s equation are possible if n(n+1) is a positive integer. This puts restriction on values of l such that:

The possible values of orbital quantum number (l) are:

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Hydrogen Atom

A hydrogen atom consists of a proton around which the electron revolves under the influence of attractive Coulomb’s force which provides the necessary centripetal force.
The Coulomb’s force of attraction between proton and electron generates potential which is purely a function of distance of separation between two. Hence this force is spherically symmetric in nature.

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Schrodinger’s eq. in spherical polar coordinate system is:

The angular part of this differential equation yields spherical harmonics as solution which is independent of r and nature of spherically symmetric potential. Hence we concentrate on solution of radial equation which is:

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The wavefunction ψ(r,θ,ϕ) must be well defined so R(ρ) must vanish for large values of r. The general solution for (2.197) is:

Substituting eq. (2.199) in (2.197) and multiplying by ρ, we have:

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From the theory of differential equations, we know that Laguerre’s differenital equation is given as::

The eqs. (2.200) and (2.201) are similar and hence L(ρ) denotes the Laguerre’s polynomial. In the equation (2.200), the quantity given as:

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Energy of State

Using expression for δ, we have:

The principal quantum number (n) defines the quantized energies of permissible states of hydrogen atom.
The negative energy of states indicates their bound nature.

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Quantum Numbers

The total wavefunction of a given state of Hydrogen atom can be expressed as the product of three functions once their dependence on different quantum numbers (n,l,m) are established.

The three quantum numbers defining the state of a hydrogen atom can assume possible values as:

Quantum Number	Symbol	Possible values
Principal Quantum Number	n	1,2,3,4,…
Orbital Quantum Number	l	0,1,2,……,(n-1)
Magnetic Quantum Number	m	0,±1, ±2, ±3,…..,l

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Principal Quantum Number

The electron in the hydrogen atom revolves around the nucleus with electrostatic force providing the necessary centripetal force.
As the electron is trapped in the potential well created by electrostatic field of nucleus, its total energy and angular momentum must remain conserved.
The possible energies of electron bound to the nucleus are negative and quantized given by:

Hence the quantized energies of the electron in the hydrogen atom are described by the principal quantum number.

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Orbital Quantum Number

The radial component of Schrodinger’s equation is expressed as

This equation is concerned with radial motion of electron that is motion towards or away from the nucleus. The term E in this equation represents total energy which is sum of kinetic and potential energies. Hence we have:

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For Schrodinger’s equation to explicitly represent radial motion of electron, the orbital motion terms must obey the condition that:

For given n, the possible values of l are:

The orbital angular momentum magnitudes are expressed for different values of orbital quantum numbers as:

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Magnetic Quantum Number (m)

The orbital quantum number signifies the quantized magnitudes of orbital angular momentum of electron.
The angular momentum of electron being a vector quantity needs direction too.
The possible directions of orbital angular momentum of electron are also quantized which are denoted by magnetic quantum number.
The component of orbital angular momentum about magnetic field along z-axis are given as:

These components denote the quantized orientation of orbital angular momentum and hence those of electron’s orbit in space.

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Stationary State of Electron

According to Bohr’s frequency rule, the radiation of frequency ν is emitted when an electron jumps from higher state m to some lower state n in the atom. This is given to be:

Let’s consider a system in which an electron moves in x-direction only. The time dependent wave function of electron in the state defined by quantum number n is expressed as:

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The expectation value of electron’s position is expressed as:

It is revealed that the position of the electron in a specific quantum state does not change with time. It implies that the electron in a quantum state does not oscillate as an electric dipole.
Hence such an electron does not emit electromagnetic radiations. It is due to this reason that these quantum states are called stationary states.

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Radiative Transitions

Let’s consider that an electron is in excited state defined by quantum number m and energy E_m. The time dependent wavefunction for this electron is represented as:

Similarly for an electron in quantum state defined by quantum number n, the wavefunction will be represented as:

The electron is non-radiating when present in any of these two states as its position will not change with time.

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Now let’s consider that electron de-excites from quantum state m to that characterized by n. The wavefunction of electron representing this process of transition can be expressed as:

Let’s consider again that electron moves along x-direction, then its expectation value of position will be:

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The first and second terms in the expression denote the expectation value of position of electron existing in the quantum states defined by m and n respectively. These expectation values are not time dependent and hence electron does not radiate in these states.
The third term denotes the expectation value of electron’s position while it is in transition from quantum state m to that defined by n. The cosine term is time dependent indicative of continuous change of position of electron.
The continuously changing position of electron denotes electric dipole oscillations which results in emission of electromagnetic radiations.
As electron can oscillate in any direction, so it behaves like a dipole antenna in that direction resulting in emission of EM radiation.

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Selection Rules

It is quite evident that when an electron de-excites, it emits EM radiations due to its electric dipole oscillations. The term in generalized form representing oscillation is:

If wavefunctions denoting different quantum states of atom are used in the above integral, then it is observed that transitions between various states are allowed only if the quantum number of states obey the condition that:

These are called selection rules for allowed atomic transitions.

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Stern Gerlach Experiment

Space quantization was explicitly demonstrated by Stern and Gerlach in 1921.
The silver atoms were produced by evaporation in an oven.
These atoms were passed through a set of collimating slits to produce a fine beam of neutral silver atoms.
The trace of the beam was recorded on the photographic plate after passing through the field.
The neutral silver atoms have odd valence electron and due to its spin, it behaves as a magnetic dipole. The dipole moment of atoms in the beam are oriented in random fashion.

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In presence of uniform magnetic field, the magnetic dipole moment of all the silver atoms align along the direction of field leading to straight line trace on the photographic plate.
In the presence of inhomogeneous field, the magnetic dipoles will suffer different force depending upon their relative orientation with respect to the field.
Classically it is expected that silver beam on passing the inhomogeneous field will show a broadened trace on the photographic plate due to continuous distribution of orientations of magnetic dipoles.

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However it was observed that in the presence of inhomogeneous magnetic field, the straight line trace splits into two, indicating two possible orientations of the magnetic dipole moment.
These two orientations arise due to the fact that the silver atom beam splits into two distinct parts corresponding to two different intrinsic spin orientations.
These observation clearly revealed that electron has intrinsic spin which has two quantized values.

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Zeeman Effect

It is a Magneto-Optical phenomenon discovered by Zeeman in 1896.
It is experimentally observed when radiating atom is free from influence of external magnetic field, the spectral lines appear as singlet.
When the radiating atom is placed in external magnetic field, the earlier observed singlet spectral lines, split into two or more components. This is referred to as Zeeman Effect.

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Experimental Set-up

The source of light (T) was placed symmetrically between pole pieces of a strong electromagnet (M).
A hole was drilled parallel to the direction of magnetic field in one of the pole pieces.
The light emitted from the source was observed by means of spectrometers S₁ and S₂ in directions parallel and perpendicular to the magnetic field respectively.

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Observations

Spectrometers were focused on certain spectral lines without applying external magnetic field. These were observed to be singlet.
When the same spectral lines were observed after applying magnetic field, they appeared to split into two or more components.
Configuration of split lines depended upon the
Direction of observation (longitudinal or transverse) w.r.t applied field.
Strength of field.

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Normal & Anomalous Zeeman Effect

Normal Zeeman Effect

With ordinary magnetic field, each spectral line is split up into:

Two components when viewed in longitudinal direction
Three components when viewed in transverse direction.
Anomalous Zeeman Effect

When weak magnetic field is applied, then each spectral line is observed to split into more than three components.

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Theory of Normal Zeeman Effect

Let’s consider an atom whose angular momentum described by orbital angular momentum quantum number l is given as:

The magnetic moment associated with angular momentum is

When this atom is placed in uniform magnetic field, then a torque is exerted by the field. This leads to precession (called Larmor’s precession) of atomic magnetic moment about the direction of external magnetic field as axis.
The interaction of orbital magnetic moment of electron with external magnetic field alters the energy of atom.

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The interaction energy is given as:

In an atom, electron undergoes transition between two states obeying the selection rule
The change in frequency of spectral line due to interaction energy of magnetic dipole is:

Hence frequencies of spectral lines observed will be:

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