Quantum Mechanical Model of Atom | Chemistry Class 11 - NEET PDF Download

Need for Quantum Mechanics

Classical mechanics, based on Newton's laws, successfully explains the motion of large objects such as falling stones, projectiles and planets. However, it fails to describe reliably the behaviour of very small particles like electrons, atoms and molecules. The main reasons are:

• Very small particles can exhibit both wave-like and particle-like behaviour, a feature not accounted for in classical mechanics.
• There is an inherent uncertainty in simultaneously knowing certain pairs of properties (for example, position and momentum) of microscopic particles.

To explain phenomena at atomic and sub-atomic scales scientists developed Quantum Mechanics, the theory that:

• describes the motion and interactions of microscopic particles (electrons, atoms, photons),
• treats matter as having dual particle-wave nature, and
• reduces to classical mechanics in the limit of large masses or large quantum numbers (correspondence principle).

The modern quantum formalism was developed in the mid-1920s. Important contributors are Werner Heisenberg and Erwin Schrödinger (both 1926). Schrödinger received the Nobel Prize in Physics in 1933 for his work.

Quick Guide to Quantum Mechanics

Features of the Quantum-Mechanical Model of the Atom

The quantum-mechanical model, founded on the Schrödinger equation, gives a consistent and predictive description of atomic structure. Its principal features are:

1. Quantized energy: Electrons bound in atoms can possess only certain discrete energy values. This quantization arises from boundary conditions imposed on wave functions for bound states.
2. Electronic energy levels: Allowed energy levels (and their separation) result from the wave-nature of electrons and from solving the Schrödinger equation for the potential produced by the nucleus.
3. Heisenberg uncertainty principle: It is impossible to determine simultaneously the exact position and exact momentum (or velocity) of an electron. Only probability distributions for such observables can be given.
4. Atomic orbitals: An orbital is a region of space described by a wave function ψ where an electron associated with a particular energy state is likely to be found. Each orbital has a definite energy; an orbital can hold a maximum of two electrons (with opposite spins).
5. Probability density: The quantity |ψ|² gives the probability density of finding an electron at a point. Integrals of |ψ|² over regions give probabilities for finding the electron in those regions.

Schrödinger Wave Equation

• The Schrödinger wave equation is a fundamental differential equation of quantum physics that describes how the quantum state (wave function) of a physical system changes in space and time.
• It is widely used in chemistry and physics to determine allowed energies and corresponding wave functions for electrons in atoms and molecules.
• Schrödinger's approach combines the wave properties of matter (de Broglie hypothesis), the classical wave equation form, and the conservation of energy to obtain a quantum wave equation.

What is the Schrödinger Wave Equation?

The Schrödinger equation is a mathematical expression that determines the wave function ψ(r, t) of a particle and thereby predicts probabilities of physical observables. Its derivation rests on three building blocks:

• the classical plane-wave form and the wave equation,
• de Broglie's relation linking momentum and wavelength for matter waves, and
• conservation of energy (kinetic + potential energy).

There are two commonly used forms:

• Time-dependent Schrödinger equation - describes the full evolution of the wave function ψ(r, t) in time and space.
• Time-independent Schrödinger equation - used for systems with time-independent potentials; it gives stationary states (energy eigenstates) and eigenvalues E.

Time-dependent Schrödinger equation (compact form):

Time-dependent Schrödinger equation in the position representation:

In the above, i is the imaginary unit, ψ is the time-dependent wave function, ħ is the reduced Planck constant, V(x) is the potential energy, and

is the Hamiltonian operator (total energy operator).

Time-independent Schrödinger equation (operator form):

For a non-relativistic particle the standard stationary form is:

Derivation (Outline)

Classical Plane-Wave Equation

A plane wave solution for a classical wave satisfies a second-order linear wave equation. The one-dimensional wave equation for a scalar amplitude y(x,t) is a second-order partial differential equation:

A sinusoidal plane progressive wave can be written as:

In these expressions A is the amplitude, T the period, φ a phase constant, and t is time. For standing waves the time and space dependence differ accordingly.

The general wave equation can be expressed in compact partial differential form:

de Broglie's Hypothesis of Matter Waves

Planck's relation links energy and frequency:

E = hν = 2πħν

Louis de Broglie proposed that every particle of momentum p has an associated wavelength λ given by:

so that

For an electron treated as a matter wave, a plane-wave form can be written for the wave function ψ. Using the de Broglie relations and expressing ψ as a sinusoidal wave gives the dependence on momentum and energy. The partial derivatives with respect to x and t of this ψ produce relations that substitute momentum and energy operators into the wave equation.

Conservation of Energy

Total mechanical energy equals kinetic plus potential energies:

Substituting the operator forms for energy and momentum into the wave equation and rearranging leads to the time-dependent Schrödinger equation for a one-dimensional particle:

The three-dimensional time-dependent Schrödinger equation for a single particle in potential V(r) is then:

On reorganisation it is commonly written as:

Equivalently, the stationary Schrödinger equation is written as

or in operator notation as Ĥψ = Eψ, where Ĥ is the Hamiltonian operator.

Hydrogen Atom and the Schrödinger Equation

Solving the time-independent Schrödinger equation for the Coulomb potential (hydrogen atom) yields quantized energy levels and associated wave functions ψ(r, θ, φ). The solutions are characterised by three independent quantum numbers:

• Principal quantum number, n - determines the main energy level (n = 1, 2, 3, ...).
• Azimuthal (orbital) quantum number, l - determines the shape of the orbital (l = 0, 1, 2, ... , n-1).
• Magnetic quantum number, m_l - determines the orientation of the orbital in space (m_l = -l, ..., 0, ..., +l).

For hydrogen and hydrogen-like (one-electron) systems the wave functions are known analytically and are called atomic orbitals. The probability of finding the electron at a point is given by |ψ|². Quantum-mechanical results for hydrogen reproduce and extend the Bohr model predictions and explain fine details of spectra that Bohr's model could not.

Challenges for Multi-Electron Atoms

• The Schrödinger equation can be solved exactly only for very simple systems (hydrogen and hydrogen-like ions). For atoms with more than one electron the equation cannot be solved exactly due to electron-electron interactions.
• Practical treatment of multi-electron atoms requires approximate methods (for example, Hartree-Fock, configuration interaction, density functional theory). Modern computers perform large scale numerical calculations based on these approximations.
• Compared with hydrogen, orbitals in multi-electron atoms are generally contracted because of increased effective nuclear charge felt by electrons. Also, orbital energies depend on both n and l (not only on n as in hydrogen).

Graphs for Various Wave Functions

Radial Wave Function, R(r) or Ψ(r)

The radial part of the hydrogenic wave function, R(r), depends only on the distance r from the nucleus. For bound states R(r) → 0 as r → ∞. The number of radial nodes of an ns orbital equals n - 1; np orbitals have n - 2 radial nodes, and so on. Nodes are points where the radial function changes sign (passes through zero). Radial plots show how R varies with r and reveal positions of nodes.

Variation of R with r.

Radial Probability Density, R²(r) or Ψ²(r)

The square of the radial function, R²(r) (or Ψ² for the radial part), gives the local radial electron density at distance r. A plot of R² versus r shows the relative electron density along a radius. For s orbitals the electron density is non-zero at the nucleus; for p, d, f orbitals the density at the nucleus is zero.

Plot of R² with r.

Radial Probability Function, 4πr²R²(r)

Because atoms are spherically symmetric, it is more useful to consider the probability of finding the electron in a thin spherical shell between r and r + dr. The shell volume is 4πr² dr. The probability of finding the electron in that shell is 4πr² R²(r) dr.

The radial probability function 4πr²R²(r) gives the probability (per unit dr) of finding the electron at distance r from the nucleus regardless of direction.

Typical radial probability distribution curves for the hydrogenic 1s, 2s and 2p orbitals illustrate the most probable radii (peaks) and the presence of nodes for higher orbitals.

Hydrogen 1s Radial Probability

Hydrogen 3s Radial Probability

Hydrogen 3p Radial Probability

Hydrogen 3d Radial Probability

Angular Wave Function (QF) and Angular Probability Density |QF|²

The total hydrogenic wave function separates into a radial part R(r) and an angular part Y(θ, φ) (often denoted by spherical harmonics). The angular wave function depends only on the quantum numbers l and m_l and is independent of n for a given orbital type. Thus all s orbitals have the same angular shape (spherical).

For an s orbital the angular part is constant (no angular dependence). For p orbitals the angular part yields two lobes (e.g., p_z along z axis). The p_x and p_y shapes are identical in form but oriented along x and y axes respectively. d and f orbitals show more complex multi-lobed angular shapes (four-lobed for many d orbitals, etc.).

Angular probability density is obtained by squaring the angular function; for s orbitals this leaves the spherical shape unchanged while for other orbitals the squared amplitude modifies relative weights of lobes.

Fig: Angular wave function.

Important Questions For Schrödinger Equation

1. What is a wave function?
Answer: Wave function is used to describe 'matter waves'. Matter waves are very small particles in motion having a wave nature - dual nature of particle and wave. Any variable property that makes up the matter waves is a wave function of the matter wave. Wave function is denoted by a symbol 'Ψ'. Amplitude, a property of a wave, is measured by following the movement of the particle with its Cartesian coordinates with respect of time. The amplitude of a wave is a wave function. The wave nature and the amplitudes are a function of coordinates and time, such that, Wave function Amplitude = Ψ = Ψ(r, t); where, 'r' is the position of the particle in terms of x, y, z directions.

2. What is meant by stationary state and what is its relevance to atom?
Answer: Stationary state is a state of a system, whose probability density given by | Ψ² | is invariant with time. In an atom, the electron is a matter wave, with quantized angular momentum, energy, etc. Movement of the electrons in their orbit is such that probability density varies only with respect to the radius and angles. The movement is akin to a stationary wave between two fixed ends and independent of time. Wave function concept of matter waves are applied to the electrons of an atom to determine its variable properties.

3. What is the physical significance of the Schrödinger wave function?
Answer: Bohr's concept of an atom is simple. However, it cannot explain the presence of multiple orbitals and the fine spectrum arising out of them. It is applicable only to the one-electron system. Schrödinger wave function has multiple unique solutions representing characteristic radius, energy, and amplitude. The probability density of the electron calculated from the wave function shows multiple orbitals with unique energy and distribution in space. Schrödinger equation could explain the presence of multiple orbitals and the fine spectrum arising out of all atoms, not necessarily hydrogen-like atoms.

4. What is the Hamilton operator used in the Schrödinger equation?
Answer: In mathematics, an operator is a rule that converts one function into another. For example, 'A' will be an operator if it can change a property f(x) into another function f(y). The Hamiltonian operator is the sum of potential and kinetic energy operators calculated over spatial coordinates. Hamiltonian operator = Ĥ = T + V = Kinetic energy + Potential energy

5. The electrons are more likely to be found:

(1) in the region a and b
(2) in the region a and c
(3) only in the region c
(4) only in the region a

Regions a and c have the maximum amplitude (Ψ) and hence the maximum probability density of electrons | Ψ² |

Answer: (2)