Quantum Mechanical Model

Introduction

The quantum mechanical model, developed by Schrödinger, Heisenberg, and others in the 1920s, replaced Bohr’s model with a more accurate description of atomic structure based on probability rather than definite orbits.

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Schrödinger Wave Equation

The Equation

$$\hat{H}\psi = E\psi$$

or in full form for a hydrogen atom:

$$\frac{\partial^2 \psi}{\partial x^2} + \frac{\partial^2 \psi}{\partial y^2} + \frac{\partial^2 \psi}{\partial z^2} + \frac{8\pi^2 m}{h^2}(E - V)\psi = 0$$

where:

• $\psi$ = wave function
• $E$ = total energy
• $V$ = potential energy
• $\hat{H}$ = Hamiltonian operator

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Wave Function (ψ)

Physical Meaning

The wave function $\psi$ itself has no direct physical meaning, but:

$$|\psi|^2 = \text{Probability density}$$

This gives the probability of finding an electron at a given point.

Key Properties

1. $\psi$ must be single-valued (one value at each point)
2. $\psi$ must be continuous (no breaks)
3. $\psi$ must be finite (no infinite values)
4. Total probability must equal 1: $\int |\psi|^2 dV = 1$

Orbitals vs Orbits

Bohr’s Orbit	Quantum Orbital
Definite circular path	3D region of probability
Electron has fixed position	Electron has probability distribution
2D picture	3D wave function

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Quantum Numbers

Four quantum numbers completely describe an electron’s state:

graph TD
    A[Quantum Numbers] --> B["n: Principal
Shell, Size, Energy"]
    A --> C["l: Azimuthal
Subshell, Shape"]
    A --> D["mₗ: Magnetic
Orbital orientation"]
    A --> E["mₛ: Spin
Electron spin"]

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1. Principal Quantum Number (n)

Definition

Describes the shell and determines the size and energy of the orbital.

Allowed Values

$$n = 1, 2, 3, 4, ... \text{ (positive integers)}$$

Significance

n	Shell	Max electrons (2n²)
1	K	2
2	L	8
3	M	18
4	N	32

• Energy: $E \propto -\frac{1}{n^2}$ (for H-like atoms)
• Size: $r \propto n^2$

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2. Azimuthal/Angular Momentum Quantum Number (l)

Definition

Describes the subshell and determines the shape of the orbital.

Allowed Values

$$l = 0, 1, 2, ..., (n-1)$$

For each n, there are n possible values of l.

Subshell Notation

l	Subshell	Shape	Number of orbitals
0	s	Spherical	1
1	p	Dumbbell	3
2	d	Double dumbbell	5
3	f	Complex	7

Angular Momentum

$$L = \sqrt{l(l+1)} \cdot \frac{h}{2\pi}$$

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3. Magnetic Quantum Number (mₗ)

Definition

Describes the orientation of the orbital in space.

Allowed Values

$$m_l = -l, -(l-1), ..., 0, ..., (l-1), +l$$

Total values = (2l + 1)

Examples

Subshell	l	mₗ values	Number of orbitals
s	0	0	1
p	1	-1, 0, +1	3
d	2	-2, -1, 0, +1, +2	5
f	3	-3, -2, -1, 0, +1, +2, +3	7

Interactive Demo: Visualize 3D Orbital Shapes

Explore the shapes of s, p, d, and f orbitals in three dimensions.

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4. Spin Quantum Number (mₛ)

Definition

Describes the intrinsic angular momentum (spin) of the electron.

Allowed Values

$$m_s = +\frac{1}{2} \text{ or } -\frac{1}{2}$$

• $+\frac{1}{2}$: spin up (↑)
• $-\frac{1}{2}$: spin down (↓)

Spin Angular Momentum

$$S = \sqrt{s(s+1)} \cdot \frac{h}{2\pi} = \sqrt{\frac{3}{4}} \cdot \frac{h}{2\pi}$$

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Summary Table

Quantum Number	Symbol	Values	Determines
Principal	n	1, 2, 3, …	Shell, size, energy
Azimuthal	l	0 to (n-1)	Subshell, shape
Magnetic	mₗ	-l to +l	Orbital orientation
Spin	mₛ	±½	Electron spin

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Counting Electrons

Maximum Electrons

• In an orbital: 2 (opposite spins)
• In a subshell: 2(2l + 1)
• In a shell: 2n²

Shell	Subshells	Total orbitals	Max electrons
K (n=1)	1s	1	2
L (n=2)	2s, 2p	1+3=4	8
M (n=3)	3s, 3p, 3d	1+3+5=9	18
N (n=4)	4s, 4p, 4d, 4f	1+3+5+7=16	32

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Quantum Number Problems

Example 1: Valid Set

Problem: Which set is valid? (n, l, mₗ, mₛ) a) (2, 2, 0, +½) b) (3, 2, -2, -½) c) (2, 1, 2, +½)

Solution:

• a) Invalid: l cannot equal n (l must be < n)
• b) Valid: n=3, l=2 (d subshell), mₗ=-2 (allowed for l=2), mₛ=-½ (correct)
• c) Invalid: mₗ=2 not allowed when l=1 (range is -1 to +1)

Example 2: Electron Count

Problem: How many electrons in an atom can have n=3, l=2?

Solution: n=3, l=2 → 3d subshell Number of orbitals = 2(2)+1 = 5 Maximum electrons = 5 × 2 = 10

Example 3: Possible Sets

Problem: How many possible sets of quantum numbers exist for n=2?

Solution: For n=2: l = 0, 1

l	mₗ	mₛ	Total
0	0	±½	2
1	-1, 0, +1	±½ each	6

Total = 8 (which equals 2n² = 2×4 = 8, correct!)

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Key Points for JEE

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Practice Problems

1. Write all possible sets of quantum numbers for 3p electrons.

2. An electron has quantum numbers n=4, l=2. What are the possible values of mₗ?

3. How many orbitals in an atom can have n=4 and mₗ=0?

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