Schrödinger equation for si and ge in hydrogen atom

Schrödinger equation all topics are cover

2. Introduction to
Schrödinger's
Equation
Schrödinger's equation is a fundamental equation in quantum
mechanics that describes the behavior of particles at the atomic and
subatomic levels.

3. Solving Schrödinger's Equation for
Hydrogen Atoms
The Schrödinger equation can be solved analytically for the hydrogen atom, which has a single proton
and a single electron. The solution provides a set of energy levels and corresponding wave functions,
which describe the probability of finding the electron at a particular location around the nucleus.
1
Step 1: Setting Up the Equation
The Schrödinger equation for the hydrogen atom is a complex differential equation that
involves the potential energy of the electron and the kinetic energy of the electron.
2
Step 2: Solving the Equation
The solution to the equation is obtained by using a method called separation of variables,
which separates the equation into two simpler equations that can be solved
independently.
3
Step 3: Obtaining the Solutions
The solutions to the equation are a set of energy levels and corresponding wave
functions, which describe the probability of finding the electron at a particular location
around the nucleus.

4. Atomic Orbitals and Quantum Numbers
Atomic orbitals are described by a set of four quantum numbers, which specify the energy, shape, and orientation of an
electron in an atom.
Principal Quantum Number (n)
Describes the energy level of an
electron. Higher values of n
correspond to higher energy levels.
Angular Momentum
Quantum Number (l)
Describes the shape of an
electron's orbital. l = 0, 1, 2, and 3
correspond to s, p, d, and f
orbitals, respectively.
Magnetic Quantum Number
(ml)
Describes the orientation of an
electron's orbital in space. It can
take on values from -l to +l,
including 0.

5. Valence Electron Energy for Silicon
(Si)
Silicon (Si) has 14 electrons, with the electronic configuration 1s² 2s² 2p⁶ 3s² 3p². The
valence electrons are those in the outermost shell, which are the 3s² 3p² electrons.
Electronic Configuration
The electronic configuration of silicon is 1s² 2s² 2p⁶ 3s² 3p².
Valence Electrons
The valence electrons are those in the outermost shell, which are the 3s² 3p²
electrons.
Valence Electron Energy
The energy of the valence electrons can be determined using the ionization
energy, which is the energy required to remove an electron from an atom.

6. Quantum Mechanical Model of the Atom
The quantum mechanical model of the atom describes the behavior of electrons in atoms using the
principles of quantum mechanics.
1
Quantized Energy Levels
Electrons in atoms can only occupy
specific energy levels, known as quantized
energy levels. The energy levels are
discrete, meaning that electrons can only
exist at specific energies.
2
Wave-Particle Duality
Electrons exhibit wave-particle duality,
meaning that they can behave as both
particles and waves. This is reflected in
the wave function, which describes the
probability of finding an electron at a
particular location.
3
Uncertainty Principle
The uncertainty principle states that it is
impossible to know both the position and
momentum of an electron with perfect
accuracy. This is a fundamental limitation
of quantum mechanics.
4
Atomic Orbitals
The wave functions for electrons in atoms
are called atomic orbitals. Each orbital
describes a specific energy level and
shape for an electron in an atom.

7. Valence Electron Energy for
Germanium (Ge)
Germanium (Ge) has 32 electrons, with the electronic configuration 1s²
2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p². The valence electrons are those in the
outermost shell, which are the 4s² 4p² electrons.
Electronic Configuration 1s² 2s² 2p⁶ 3s² 3p⁶ 4s² 3d¹⁰ 4p²
Valence Electrons 4s² 4p²
Valence Electron Energy The energy of the valence
electrons can be determined
using the ionization energy,
which is the energy required
to remove an electron from an
atom.

8. Comparison of Valence
Electron Energies
The valence electron energies of silicon and germanium are similar,
but not identical.
Silicon (Si)
Silicon has a slightly higher
ionization energy than
germanium, indicating that its
valence electrons are slightly
more tightly bound to the
nucleus.
Germanium (Ge)
Germanium has a slightly lower
ionization energy than silicon,
indicating that its valence
electrons are slightly less tightly
bound to the nucleus.

9. Conclusion and Applications
The understanding of valence electron energy plays a crucial role in
various fields, including materials science, chemistry, and
electronics.
Semiconductors
Silicon and germanium are
semiconductors, meaning
that they have a
conductivity between that
of a conductor and an
insulator. This property is
essential for the
development of transistors
and other electronic
devices.
Solar Cells
Silicon is used in solar
cells to convert sunlight
into electricity. The valence
electron energy levels of
silicon determine its ability
to absorb sunlight and
release electrons.