### 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*(2*ed*) 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>Loads an ISW initial Gaussian wave packet with no initial average momentum._{0}= 0:ISW Narrow Gaussian <p>Loads an ISW initial Gaussian wave packet with an initial average momentum._{0}= 80 π/a:

## 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.

### For Teachers

### Translations

### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support full-screen? | Yes. Chrome/Opera No. Firefox/ Samsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

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

### end faq

### Sample Learning Goals

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### For Teachers

## Bound Eigenstate Superposition JavaScript Simulation Applet HTML5

### Instructions

#### Combo Box and Functions

#### Toggling Full Screen

#### Play/Pause, Step and Reset Buttons

Research

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