Mathematics · Quantum Mathematics
Infinite Square Well Energy Calculator
Calculate a particle's quantized energy level in a one-dimensional infinite square well.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Convert mass=1mₑ=9.1093837139e-31 kg and width=1 nm=1e-9 m.
- Use n²=1 and L²=1e-18.
- E_1=6.024667386653732e-20 J=0.37603016164407077 eV.
Understand Infinite well energy
One idea, three depths
Choose how deeply to explain Infinite well energy
Infinite well energy: Calculate a particle's quantized energy level in a one-dimensional infinite square well.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Infinite well energy to answer this question: calculate a particle's quantized energy level in a one-dimensional infinite square well? Enter Quantum number n, Mass in electron masses, Well width L; the calculator shows Energy Eₙ. For example: An electron in a 1 nm well has ground-state energy about 0.376 eV. The answer tells you Energy Eₙ.
Age 15Explain it to a 15-year-oldConnect it to the formula
Boundary conditions allow only standing waves that fit the well, producing discrete energies proportional to n² and inversely proportional to L². The rule is Eₙ=n²π²ℏ²/(2mL²). Its input values are Quantum number n, Mass in electron masses (mₑ), Well width L (nm), and the main result is Energy Eₙ. For example: An electron in a 1 nm well has ground-state energy about 0.376 eV.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated infinite well energy relation over the valid mixed integer and real-number domain stated below. The implemented relation is Eₙ=n²π²ℏ²/(2mL²), evaluated from Quantum number n, Mass in electron masses (mₑ), Well width L (nm) to produce Energy Eₙ. Boundary conditions allow only standing waves that fit the well, producing discrete energies proportional to n² and inversely proportional to L². This ideal model assumes infinitely high walls and a non-relativistic particle; real wells have finite barriers.
Inputs and valid domain
- Quantum number n must be an integer, at least 1.
- Mass in electron masses must be a finite real number, at least 0 in mₑ.
- Well width L must be a finite real number, at least 0 in nm.
Important boundary: This ideal model assumes infinitely high walls and a non-relativistic particle; real wells have finite barriers.
The formula
Eₙ=n²π²ℏ²/(2mL²)
How the calculator works through it
It substitutes Quantum number n, Mass in electron masses, Well width L into the formula and exposes every numerical step above. The main output is Energy Eₙ, accompanied by Energy in electronvolts, Mass in kilograms, Well width in metres.
Read the result correctly
The Energy Eₙ is the direct answer to “calculate a particle's quantized energy level in a one-dimensional infinite square well.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
An electron in a 1 nm well has ground-state energy about 0.376 eV.
Where this model stops being reliable
This ideal model assumes infinitely high walls and a non-relativistic particle; real wells have finite barriers.
Learn it by changing one value
Begin with the worked example, then change one value while keeping the others fixed. Compare the new result and calculation steps to identify which part of the formula changed.
Dictionary terms behind this calculator
Before studying the codeWhat you should know firstUse the calculator immediately, or check the foundations before reading the implementation.
These foundations help you understand why Infinite well energy works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Infinite well energy uses Eₙ=n²π²ℏ²/(2mL²). You need to recognise what each side represents before substituting the stated inputs or rearranging the relationship.
Review this foundation about 4 min
Strong support
- Probability and normalised outcomes
Probability interpretation is needed to connect the Infinite well energy mathematics to measurable outcomes.
Review this foundation about 6 min
Optional enrichment
- Complex amplitudes
Complex-number notation gives deeper context for amplitudes and phase relationships related to Infinite well energy.
Review this foundation about 7 min
Mathematics → algorithm → program
Implement this calculation in code
These are direct reference implementations of the calculator's principal relationship and first output. They run locally and include a small known-answer check where the language supports it.
Algorithm
- Read Quantum number n, Mass in electron masses, Well width L.
- Evaluate the principal relationship: Eₙ=n²π²ℏ²/(2mL²).
- Return Energy Eₙ and check the domain conditions described above.
Python
from math import *
def infinite_well_energy(n, a, b) -> float:
return ((((n * n) * (pi * pi)) * (1.054571817e-34 * 1.054571817e-34)) / (2.0 * ((a * 9.1093837139e-31) * ((b * 1e-9) * (b * 1e-9)))))
assert abs(infinite_well_energy(1, 1, 1) - 6.024667386653732e-20) < 1e-6 * max(1.0, abs(6.024667386653732e-20))
C
#include <assert.h>
#include <math.h>
double infinite_well_energy(double n, double a, double b) {
return ((((n * n) * (3.141592653589793 * 3.141592653589793)) * (1.054571817e-34 * 1.054571817e-34)) / (2.0 * ((a * 9.1093837139e-31) * ((b * 1e-9) * (b * 1e-9)))));
}
int main(void) {
const double expected = 6.024667386653732e-20;
const double actual = infinite_well_energy(1, 1, 1);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double infinite_well_energy(double n, double a, double b) {
return ((((n * n) * (std::numbers::pi * std::numbers::pi)) * (1.054571817e-34 * 1.054571817e-34)) / (2.0 * ((a * 9.1093837139e-31) * ((b * 1e-9) * (b * 1e-9)))));
}
int main() {
constexpr double expected = 6.024667386653732e-20;
const double actual = infinite_well_energy(1, 1, 1);
assert(std::fabs(actual - expected) < 1e-6 * std::fmax(1.0, std::fabs(expected)));
}
Linux x86-64 assembly
x86-64 NASM · System V ABI · Linux · SSE2 with libm where required
; double infinite_well_energy(double n, double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
global infinite_well_energy
section .text
infinite_well_energy:
push rbp
mov rbp, rsp
sub rsp, 192
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
movsd [rbp-24], xmm2
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-8]
movsd [rbp-56], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-72], xmm0
mov rax, 0x400921fb54442d18
movq xmm0, rax
movsd [rbp-80], xmm0
movsd xmm0, [rbp-72]
mulsd xmm0, [rbp-80]
movsd [rbp-64], xmm0
movsd xmm0, [rbp-56]
mulsd xmm0, [rbp-64]
movsd [rbp-48], xmm0
mov rax, 0x38e185a7054e4c91
movq xmm0, rax
movsd [rbp-96], xmm0
mov rax, 0x38e185a7054e4c91
movq xmm0, rax
movsd [rbp-104], xmm0
movsd xmm0, [rbp-96]
mulsd xmm0, [rbp-104]
movsd [rbp-88], xmm0
movsd xmm0, [rbp-48]
mulsd xmm0, [rbp-88]
movsd [rbp-40], xmm0
mov rax, 0x4000000000000000
movq xmm0, rax
movsd [rbp-120], xmm0
mov rax, 0x39b279dcc922bcd9
movq xmm0, rax
movsd [rbp-144], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-144]
movsd [rbp-136], xmm0
mov rax, 0x3e112e0be826d695
movq xmm0, rax
movsd [rbp-168], xmm0
movsd xmm0, [rbp-24]
mulsd xmm0, [rbp-168]
movsd [rbp-160], xmm0
mov rax, 0x3e112e0be826d695
movq xmm0, rax
movsd [rbp-184], xmm0
movsd xmm0, [rbp-24]
mulsd xmm0, [rbp-184]
movsd [rbp-176], xmm0
movsd xmm0, [rbp-160]
mulsd xmm0, [rbp-176]
movsd [rbp-152], xmm0
movsd xmm0, [rbp-136]
mulsd xmm0, [rbp-152]
movsd [rbp-128], xmm0
movsd xmm0, [rbp-120]
mulsd xmm0, [rbp-128]
movsd [rbp-112], xmm0
movsd xmm0, [rbp-40]
divsd xmm0, [rbp-112]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
leave
ret
MATLAB
function result = infinite_well_energy(n, a, b)
result = ((((n * n) * (pi * pi)) * (1.054571817e-34 * 1.054571817e-34)) / (2.0 * ((a * 9.1093837139e-31) * ((b * 1e-9) * (b * 1e-9)))));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[n_, a_, b_] := ((((n * n) * (Pi * Pi)) * (1.054571817e-34 * 1.054571817e-34)) / (2.0 * ((a * 9.1093837139e-31) * ((b * 1e-9) * (b * 1e-9)))));
Continue in mathematical software
The downloaded file includes your current inputs and first calculated result. It is created locally.
Floating-point answers can differ slightly by language, compiler and processor. Compare within a suitable tolerance rather than assuming every decimal representation will be identical.
Supporting sourcesAcademic referencesPrimary standards, textbooks and complete citations
Standards, reading and academic references
Use the calculator as the worked interaction, then consult the primary standards and academic textbooks listed below. MW SysArc links to the original sources; the explanation on this page is original and does not reproduce them.
University Physics Volume 3
Read OpenStax University Physics: Quantum MechanicsCite this book
- APA 7
- Ling, S. J., Sanny, J., & Moebs, W. (2016). University physics volume 3. OpenStax. https://openstax.org/books/university-physics-volume-3/pages/1-introduction
- MLA 9
- Ling, Samuel J., et al. University Physics Volume 3. OpenStax, 2016, https://openstax.org/books/university-physics-volume-3/pages/1-introduction.
- Chicago author-date
- Ling, Samuel J., Jeff Sanny, and William Moebs. 2016. University Physics Volume 3. Houston, TX: OpenStax. https://openstax.org/books/university-physics-volume-3/pages/1-introduction.
OpenStax entries are free to read online. Follow the licence shown on each linked source before redistributing or adapting its content.
Reuse the page responsiblyCite this pageAPA, MLA, Chicago, Harvard, BibTeX and RIS
These formats cite this calculator page itself. They are separate from the academic references above, which support the mathematical method and terminology.
APA 7
MW SysArc. (2026, July 21). Infinite Square Well Energy Calculator. MW SysArc Tools. https://math.mwsysarc.com/quantum-mathematics/infinite-square-well-energy
MLA 9
MW SysArc. “Infinite Square Well Energy Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/quantum-mathematics/infinite-square-well-energy. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Infinite Square Well Energy Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/quantum-mathematics/infinite-square-well-energy.
Harvard
MW SysArc (2026) ‘Infinite Square Well Energy Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/quantum-mathematics/infinite-square-well-energy (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_infinite_well_energy_2026,
author = {{MW SysArc}},
title = {Infinite Square Well Energy Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/quantum-mathematics/infinite-square-well-energy},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Infinite Square Well Energy Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/quantum-mathematics/infinite-square-well-energy
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Infinite well energy do?
Calculate a particle's quantized energy level in a one-dimensional infinite square well.
How does the Infinite well energy work?
The calculator applies Eₙ=n²π²ℏ²/(2mL²). Boundary conditions allow only standing waves that fit the well, producing discrete energies proportional to n² and inversely proportional to L².
What can I learn from the Infinite well energy?
It connects the mathematical rule to your chosen numbers and shows each calculation step. Change one input at a time to see how the result responds.
Does MW SysArc receive or store what I enter?
No. The calculation runs locally in your browser. MW SysArc does not receive or store your calculation inputs.
How should I use the result?
Use the steps to understand the method, then verify important school or professional work using the notation and rounding rules required in your setting.
Last reviewed . Calculations tested .