General Oscillatory Wave Function Equation, Derivation & Rearrangements

Core idea

Overview

Inside the well, the Schrödinger equation gives a sine-and-cosine combination because the particle energy exceeds the local potential.

When to use: Use this when the wavefunction must be matched across a finite barrier or finite well.

Why it matters: The tunneling picture explains why wavefunctions oscillate in allowed regions and decay exponentially in forbidden regions.

Walkthrough

Derivation

Derivation of the Oscillatory Wave Function

The wave function is derived by solving the time-independent Schrödinger equation for a region where the particle energy E is greater than the potential energy V.

The particle is in a region of constant potential V.
The energy of the particle E is greater than the potential V (E > V).
The particle is described by the time-independent Schrödinger equation: -( $ℏ^{2}$ / 2m) * ( $d^{2}$ $ψ$ / $d x^{2}$ ) + V $ψ$ = E $ψ$ .

1

Rearrange the Schrödinger equation

Isolating the second derivative term by moving V to the other side and dividing by the kinetic energy constant.

\frac{d ^{2} ψ}{d x ^{2}} = - \frac{2 m ( E - V )}{ℏ ^{2}} ψ

2

Define the wavenumber k

Since E > V, the term (E-V) is positive, allowing us to define k as a real constant such that $k^{2}$ = 2m(E-V)/ $ℏ^{2}$ .

k = \frac{2 m ( E - V )}{ℏ ^{2}}

3

Solve the second-order linear differential equation

This is a standard homogeneous second-order linear differential equation with constant coefficients, which has the general solution form involving sinusoidal functions.

\frac{d ^{2} ψ}{d x ^{2}} + k^{2} ψ = 0

4

State the general solution

The linear combination of sine and cosine functions forms the complete general solution to the differential equation.

ψ (x) = A_{1} sin (k x) + A_{2} cos (k x)

Result

ψ (x) = A_{1} sin (k x) + A_{2} cos (k x)

Source: Engineering LibreTexts, finite square well and tunneling-barrier notes

Free formulas

Rearrangements

Solve for $_{s} t a t u s$

b l oc k e d_{n} o n_{i} n v er t ib l e

Solve for reason

T h ee q u a t i o n co n t ain s tw o in d e p e n d e n t v a r iab l es (A 1 an d A 2) a ssoc ia t e d w i t h d i f f er e n tt r i g o n o m e t r i c f u n c t i o n so f x, mak in g i t im p oss ib l e t o i so l a t e A 1 or A 2 w i t h o u t k n o w in g t h e v a l u eso f t h eo t h er or ha v in g b o u n d a r y co n d i t i o n co n s t r ain t s . I so l a t in g x i ss imi l a r l y im p oss ib l e d u e t o t h e t r an sce n d e n t a l na t u r eo f t h es u m o f a s in e an d cos in e f u n c t i o n .

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Imagine a vibrating guitar string where the displacement is a combination of two basic shapes: one that starts at zero (sine) and one that starts at a peak (cosine). Geometrically, this equation represents a smooth, repeating wave that stays within a specific height. In the context of a potential well, this wave fills the 'allowed' space, bouncing back and forth between the walls to create a steady pattern of probability where the particle is likely to be found.

ψ_{2} (x)

Wave function in the middle region

The mathematical 'cloud' that describes the state of the particle; its square tells you the probability of finding the particle at position x.

A_{1}, A_{2}

Amplitude coefficients

Weighting factors that determine how much 'sine-character' versus 'cosine-character' the wave has to satisfy boundary conditions.

k

Wavenumber

The 'spatial frequency' of the wave; a higher k means the wave wiggles more rapidly over a short distance, indicating higher kinetic energy.

x

Position

The specific horizontal coordinate within the allowed region or potential well.

Signs and relationships

+ (Addition of terms): This represents the principle of superposition, allowing any possible oscillatory state to be constructed by summing the fundamental odd (sine) and even (cosine) solutions.
kx (Argument): The product of wavenumber and position ensures the wave repeats periodically; as x increases, the phase of the sine and cosine cycles forward.

Free study cues

Insight

Canonical usage

This equation describes the wave function in a region where the potential energy is less than the total energy, leading to oscillatory behavior. The constants A1 and A2 are determined by boundary conditions.

Common confusion

Students may forget that the wave function itself has units, which are crucial for ensuring the probability density is dimensionless.

Dimension note

While the wave function itself is not dimensionless, the probability density | $ψ$ |^2 is dimensionless, representing the probability per unit length.

Unit systems

$k$ m^-1 · The wave number k is related to the particle's energy (E) and potential energy (V) by k = sqrt(2m(E-V))/ħ, where m is mass and ħ is the reduced Planck constant. Thus, k has units of inverse length.

$x$ m · Represents position, typically measured in meters in SI.

$ψ_{2} (x)$ m^-0.5 · The wave function itself has units that ensure the probability density (|\psi|^2) is dimensionless. In one dimension, this means \psi has units of inverse square root of length.

$A_{1}$ m^-0.5 · Amplitude constant, must have the same units as \psi_2(x) to satisfy the equation.

$A_{2}$ m^-0.5 · Amplitude constant, must have the same units as \psi_2(x) to satisfy the equation.

One free problem

Practice Problem

What happens to the form of the wavefunction when a particle moves from a classically allowed region (E > V) to a classically forbidden region (E < V)?

Solve for: $ψ_{2} (x)$

Hint: Consider the sign of the kinetic energy term in the Schrödinger equation.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In writing the middle-region wavefunction for a finite square well, General Oscillatory Wave Function is used to calculate \psi_2(x) from the measured values. The result matters because it helps size components, compare operating conditions, or check a design margin.

Study smarter

Tips

Inside the classically allowed region you get sine and cosine solutions.
In the forbidden region the physically acceptable branch is exponential decay.
Barrier width matters exponentially, not linearly.
Whenever E > V in a constant-potential region, expect trig functions.

Avoid these traps

Common Mistakes

Using an oscillatory solution where the energy is below the barrier.
Forgetting to match both the wavefunction and its derivative at the boundaries.
Underestimating how quickly the tunneling signal drops with barrier width.
Using exponentials in the allowed region.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The wave function is derived by solving the time-independent Schrödinger equation for a region where the particle energy E is greater than the potential energy V.

Use this when the wavefunction must be matched across a finite barrier or finite well.

The tunneling picture explains why wavefunctions oscillate in allowed regions and decay exponentially in forbidden regions.

Using an oscillatory solution where the energy is below the barrier. Forgetting to match both the wavefunction and its derivative at the boundaries. Underestimating how quickly the tunneling signal drops with barrier width. Using exponentials in the allowed region.

In writing the middle-region wavefunction for a finite square well, General Oscillatory Wave Function is used to calculate \psi_2(x) from the measured values. The result matters because it helps size components, compare operating conditions, or check a design margin.

Inside the classically allowed region you get sine and cosine solutions. In the forbidden region the physically acceptable branch is exponential decay. Barrier width matters exponentially, not linearly. Whenever E > V in a constant-potential region, expect trig functions.

References

Sources

Engineering LibreTexts, finite square well and tunneling-barrier notes, accessed 2026-04-09
Peverati, The Live Textbook of Physical Chemistry 2, quantum weirdness/tunneling section, accessed 2026-04-09
Engineering LibreTexts, field enhanced emission and tunnelling effects, accessed 2026-04-09
NIST CODATA
IUPAC Gold Book
Griffiths, David J. (2018). Introduction to Quantum Mechanics (3rd ed.). Cambridge University Press.
Griffiths, David J. Introduction to Quantum Mechanics
Liboff, Richard L. Introductory Quantum Mechanics

General Oscillatory Wave Function