Physical Chemistry
Universitatis Chemia

Select your language

  • Arabic (اللغة العربية)
  • Deutsch (Deutschland)
  • Français (France)
  • Español (España)

Login Form

  • Forgot your password?
  • Forgot your username?

General Chemistry

  • Fundamentals of chemistry
  • The atom, molecules and ions
  • The chemical compounds
  • The chemical reactions
  • Quantum Theory
  • Periodic properties
  • Chemical bonding I
  • Chemical bonding II
  • Thermochemistry
  • Acid-base equilibrium
  • Solubility equilibrium
  • Nuclear chemistry

Thermodynamics

  • Introduction to Thermodynamics
  • First Principle | Thermodynamics
  • Second Principle | Thermodynamics
  • Material equilibrium
  • Equilibrium in one-component systems
  • Normal thermodynamic reaction functions
  • Ideal solutions
  • Electrolyte solutions
  • Real solutions

Quantum Mechanic

  • Principles and postulates | Quantum Mechanics
  • The particle in a box | Quantum Mechanics
  • Harmonic oscillator | Quantum mechanics
  • Angular momentum | Quantum mechanics
  • Hydrogenoid atom | Quantum mechanics
  • Helioid atoms | Quantum Mechanics
  • Polyelectronic atoms : Quantum mechanics

Spectroscopy

  • Introduction to spectroscopy
  • Rotation vibration in diatomic
  • Rotational and vibrational spectra in diatomics

Pysical Chemistry

  • Chemical kinetics
  • Statistical thermodynamics
  • Kinetic theory of gases
  • Problems | Kinetic theory of gases
  • Transport phenomena

We have 160 guests and no members online

  1. You are here:  
  2. Home
  3. Harmonic oscillator | Quantum mechanics

Harmonic oscillator | Quantum mechanics

One-dimensional harmonic oscillator (part 1)

Details
Written by: Germán Fernández
Category: Harmonic oscillator | Quantum mechanics
Published: 25 June 2012
Hits: 552

We set up the Schödinger equation:

 

 

The harmonic oscillator is subjected to a potential of the type: \(V(x)=\frac{1}{2}kx^2\)

 

 

Solving for the derivative of higher degree, equation (2) can be written:

 

 

Read more: One-dimensional harmonic oscillator (part 1)

Harmonic oscillator

Details
Written by: Germán Fernández
Category: Harmonic oscillator | Quantum mechanics
Published: 26 June 2012
Hits: 531

The harmonic oscillator is a physical system in which a mass oscillates around an equilibrium position subjected to an elastic potential \(V(x)=\frac{1}{2}kx^2\). The oscillation of the atoms in a molecule can be studied using the quantum version of this model. Solving the Schrödinger equation gives us the energy levels and the wave function.

 

 

One-dimensional harmonic oscillator (part 2)

Details
Written by: Germán Fernández
Category: Harmonic oscillator | Quantum mechanics
Published: 27 June 2012
Hits: 513

The differential equation (16) has a solution of the type:

 

 
 
 
We calculate the first and second derivatives of (17):
 
 
 
 
Substituting (17) and (19) in (16) and simplifying
 
 

Read more: One-dimensional harmonic oscillator (part 2)

Hermite polynomials

Details
Written by: Germán Fernández
Category: Harmonic oscillator | Quantum mechanics
Published: 04 October 2012
Hits: 482

Some expressions that allow us to calculate the Hermite polynomials are: \begin{equation}\label{ec-34} H_v(\xi)=(2\xi)^vv(v-1)(2\xi)^{v- 2}+\frac{v(v-1)(v-2)(v-3)}{2}(2\xi)^{v-4}+...... \end{equation} The Equation (\ref{ec-34}) can be written more compactly as: \begin{equation}\label{ec-35} H_v(\xi)=\sum_{k=0}^{v}(- 1)^k\frac{v!}{k!(v-2k)!}(2\xi)^{v-2k} \end{equation}

Read more: Hermite polynomials

Wave Function Normalization

Details
Written by: Germán Fernández
Category: Harmonic oscillator | Quantum mechanics
Published: 04 October 2012
Hits: 683

The wave functions of the harmonic oscillator are given by the equation (\ref{ec17}), where N is the normalization constant, which we can calculate with the following equation: \begin{equation}\label{ec-43} \int_ {-\infty}^{+\infty}\Psi_{v}^{\ast}(x)\Psi_{v}(x)dx=1 \end{equation} The normalization of the wave function for a state general $v$ gives us the following result: \begin{equation} N_v =\left(2^vv!\right)^{-1/2}\left(\frac{\beta}{\pi}\right) ^{1/4} \end{equation}

Harmonic Oscillator Problems

Details
Written by: Germán Fernández
Category: Harmonic oscillator | Quantum mechanics
Published: 04 October 2012
Hits: 510
  1. For the ground state of the one-dimensional harmonic oscillator, find the average value of the kinetic and potential energies. Check that <T>=<V> in this case. Find the most probable position of the particle for said wave function and verify that the uncertainty principle is satisfied. Data:$\int_{0}^{\infty}x^2 e^{-\alpha x^2}dx=\pi^{1/2}/4\alpha^{3/2}$

     

  2. Using the Rodrigues formula, generate the polynomials with $v=0,1,2,3....$

     

  3. The three-dimensional harmonic oscillator has the potential energy function $V(x,yz)=k_x \frac{x^2}{2} + k_y \frac{y^2}{2} + k_z \frac{z^2} {2}$, where $k_x$, $k_y$, and $k_z$ are three force constants. Using the method of separation of variables, find the eigenfunctions and the eigenvalues of the energy of said oscillator.

    Read more: Harmonic Oscillator Problems

  • Privacy Policy
  • Legal Notice
  • Cookies Policy