About

The Superposition Principle

The fundamental building blocks of one-dimensional quantum mechanics are energy eigenfunctions ψ_n(x) and energy eigenvalues E_n. For a given potential energy function V(x) and boundary conditions, energy eigenfunctions can be determined either analytically or numerically. Most of the time in quantum mechanics these energy eigenfunctions are determined in position space. Once these energy eigenstates are determined, more interesting quantum-mechanical wave functions Ψ(x,t) can be studied by applying the superposition principle

where the expansion coefficients c_n satisfy Σ c_n|² = 1. Depending on how many of the coefficients c_n are non-zero, one may have an energy eigenstate, a two-state superposition, or even an initially localized (usually Gaussian shaped) wave packet.

Any complete orthonormal set of eigenfunctions can be used to construct the wave function Ψ(x,t). This simulation uses the superposition principle to construct and display a time-dependent wave function using either infinite square well (ISW) or simple harmonic oscillator (SHO) eigenfunctions.

Units

Although the metric (MKS) system of units has become the standard international system of units, it is not well suited for computation if the quantities being computed are very large or very small. Quantum phenomena occurs on the microscopic scale at very fast times and computations are usually done using an atomic system of units in which the reduced Plank's constant ħ, the Bohr radius a_o, and the mass of the electron m are set equal to unity. The one-dimensional time independent Schrödinger equation in these units is:

In atomic units, one unit time is 2.42×10^-17 seconds, one unit of distance is 5.29×10^-11 meters, and one unit of energy is 4.36×10^-18 Joules. This simulation models a particle with the mass of an electron using these atomic units.

References:

• For an excellent tutorial on energy eigenfunction shape and the relationship to the potential energy function, see: A. P. French and E. F. Taylor, Qualitative plots of bound state wave functions, Am. J. Phys. 39, 961-962 (1971).
• An Introduction to Quantum Mechanics (2ed) by David J. Griffiths page 28.

Credits:

The Bound Energy Eigenstate Superposition JavaScript simulation was developed by Wolfgang Chrsitian using the Easy Java/JavaScript Simulations (EjsS) modeling tool. You can examine and modify this simulation if you have EjsS version 5.2 or above installed by importing the model's zip archive into EjsS. Information about EjsS is available at: <http://www.um.es/fem/Ejs/> and in the OSP ComPADRE collection <http://www.compadre.org/OSP/>.

The following Options cannot work on SHO (but works if system=ISO):

-"SHO squeezed"

-"SHO squeezed wide"

-"SHO boosted coherent"

-"SHO boosted squeezed"

-"SHO shifted ground state"

-"SHO shifted excited state"

Infinite Square Well Exercises

[Screen shot of an ISW superposition state that as it hits one side of the well.]

Pre-set Demonstrations

You can access the pre-set initial states for the infinite square well (ISW) via the textbox on the lower-left-hand side of the main simulation panel.  These show:

• ISW Two State: Loads an ISW two-state superposition (equal mix of ground state and first-excited state).
• ISW Gaussian: Loads an ISW initial Gaussian wave packet with no initial average momentum.
• ISW Narrow Gaussian <p>₀ = 0: Loads an ISW initial Gaussian wave packet with no initial average momentum.
• ISW Narrow Gaussian <p>₀ = 80 π/a: Loads an ISW initial Gaussian wave packet with an initial average momentum.

Exercises

1. Given the default infinite square well width of a = 1.57 = π/2, what are the ground-state and first-excited-state energies?  Recall that we have scaled the problem such that ħ = m = 1.

2. Since the time dependence of energy eigenstates is just, e^-iEnt/ħ, how long does it take the ground state and the first-excited state to evolve in time back to their t = 0 values? In other words, with what period do these states oscillate with time? Once you have these values compare them to each other.

3. Now select the ISW Two State and set the dt to the ground state period divided by 10.  The text field can accept simple mathematical operations such as /, *, pi, etc. Single step through the simulation and see if the wave function indeed has the same period as you calculated in Question 2.  Now set the dt to the ground state period divided by 3 and run the simulation. Single step through the simulation and describe the the wave function at these times. Why does this occur? What odes this result mean for the period of expectation values of x, <x>?

4. Now select one of the ISW Gaussian wave packets (ISW Gaussian, ISW Narrow Gaussian <p>0 = 0, ISW Narrow Gaussian <p>0 = 80 π/a) from the drop down menu. First look at all three wave packets. Describe the similarities and differences between the packets' initial shapes. Choose one packet and set the dt to the ground state period divided by 100 and run the simulation. Describe the motion of each wave packet. Which packet initially behaves like a particle in a classical infinite square well?

5. Now set the dt to the ground state period divided by 10 and single step through the simulation. What do you notice about the wave function at these times? Now set dt to the ground state period divided by 4, then 3 and single step through the simulation. Again describe what do you notice about the wave function at these times.

Infinite Square Well Eigenfunctions

[Screen shot of the energy eigenfunction describing the ISW first-excited state.]

The infinite square well (ISW) is an idealized model consisting of a point mass m inside an infinitely deep well of width a.  According to quantum mechanics, the energy eigenfunctions ψ_n(x) for a symmetric ISW are simple sinusoidal functions
 

.

The corresponding energy eigenvalues E_n scale as the principal quantum number n squared

 

.

Simple Harmonic Oscillator Eigenfunctions

[Screen shot of the SHO ground state.]

A simple harmonic oscillator (SHO) with a potential energy V(x) = ½mω²x²   has energy eigenfunctions ψ_n(x) that are expressed in terms of a Gaussian times a Hermite polynomial H_n(x)
 

 

The angular frequency ω=(K/m)^½ is that of a classical mass on a spring with spring constant K.  Substituting these SHO eigenfunctions ψ_n(x) into the time-independent Schrödinger equation shows that they have energy eigenvalues E_n that scale as the principal quantum number n
 

.

 

Note that unlike the infinite square well model, the simple harmonic oscillator energy eigenvalues are evenly spaced.

 

In order to compare ISW and SHO eigenfunctions, the spring constant is chosen so that the ground state energy eigenvalues of these two systems are equal.

 

Translations

Code	Language		Translator	Run

Credits

Wolfgang Christian; Mario Belloni; Dieter Roess; Fremont Teng; Loo Kang Wee

Sample Learning Goals

[text]

For Teachers

Bound Eigenstate Superposition JavaScript Simulation Applet HTML5

Instructions

Combo Box and Functions

Toggling Full Screen

Play/Pause, Step and Reset Buttons

Research

[text]

Video

[text]

 Version:

Other Resources

[text]

end faq

{accordionfaq faqid=accordion4 faqclass="lightnessfaq defaulticon headerbackground headerborder contentbackground contentborder round5"}