Mathematics · Quantum Mathematics
Hydrogen Energy Transition Calculator
Calculate hydrogen energy levels and the photon energy and wavelength between two principal quantum numbers.
Inputs and results stay in this browser. Change one value at a time to explore the relationship.
Calculation steps
- Initial energy=−13.6÷3²=-1.511111111111111 eV; final energy=−13.6÷2²=-3.4 eV.
- Signed change=-3.4−(-1.511111111111111)=-1.8888888888888888 eV, so this is emission.
- Photon wavelength=hc/|ΔE|=6.563869328816484e-7 m=656.3869328816485 nm.
Understand Hydrogen transition
One idea, three depths
Choose how deeply to explain Hydrogen transition
Hydrogen transition: Calculate hydrogen energy levels and the photon energy and wavelength between two principal quantum numbers.
Age 5Explain it to a 5-year-oldStart with a picture
Imagine using Hydrogen transition to answer this question: calculate hydrogen energy levels and the photon energy and wavelength between two principal quantum numbers? Enter Initial level nᵢ and Final level n_f; the calculator shows Photon energy. For example: The n=3 to n=2 transition releases about 1.89 eV, corresponding to roughly 656 nm red light. The answer tells you Photon energy.
Age 15Explain it to a 15-year-oldConnect it to the formula
Hydrogen's allowed bound-state energies are discrete, so transitions emit or absorb photons with exactly the energy difference between levels. The rule is Eₙ=−13.6 eV/n²; |ΔE|=hc/λ. Its input values are Initial level nᵢ, Final level n_f, and the main result is Photon energy. For example: The n=3 to n=2 transition releases about 1.89 eV, corresponding to roughly 656 nm red light.
CollegeExplain it at college levelState the model precisely
This calculator evaluates the stated hydrogen transition relation over the valid integer domain stated below. The implemented relation is Eₙ=−13.6 eV/n²; |ΔE|=hc/λ, evaluated from Initial level nᵢ, Final level n_f to produce Photon energy. Hydrogen's allowed bound-state energies are discrete, so transitions emit or absorb photons with exactly the energy difference between levels. This simplified formula is for hydrogen; multi-electron atoms require a different model.
Inputs and valid domain
- Initial level nᵢ must be an integer, at least 1.
- Final level n_f must be an integer, at least 1.
Important boundary: This simplified formula is for hydrogen; multi-electron atoms require a different model.
The formula
Eₙ=−13.6 eV/n²; |ΔE|=hc/λ
How the calculator works through it
It substitutes Initial level nᵢ, Final level n_f into the formula and exposes every numerical step above. The main output is Photon energy, accompanied by Photon wavelength, Wavelength in nanometres, Signed level change, Emission (−1) or absorption (+1).
Read the result correctly
The Photon energy is the direct answer to “calculate hydrogen energy levels and the photon energy and wavelength between two principal quantum numbers.” Read it with the units shown beside the inputs; a sign, angle, percentage or rate changes what the number means.
A worked check
The n=3 to n=2 transition releases about 1.89 eV, corresponding to roughly 656 nm red light.
Where this model stops being reliable
This simplified formula is for hydrogen; multi-electron atoms require a different model.
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 Hydrogen transition works. They never block the calculator, and “optional” means useful context rather than a hidden requirement.
Hard requirements
- Reading formulas and substituting values
Hydrogen transition uses Eₙ=−13.6 eV/n²; |ΔE|=hc/λ. 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 Hydrogen transition 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 Hydrogen transition.
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 Initial level nᵢ, Final level n_f.
- Evaluate the principal relationship: Eₙ=−13.6 eV/n²; |ΔE|=hc/λ.
- Return Photon energy and check the domain conditions described above.
Python
from math import *
def hydrogen_transition(a, b) -> float:
return fabs(((-13.6 / (b * b)) - (-13.6 / (a * a))))
assert abs(hydrogen_transition(3, 2) - 1.8888888888888888) < 1e-6 * max(1.0, abs(1.8888888888888888))
C
#include <assert.h>
#include <math.h>
double hydrogen_transition(double a, double b) {
return fabs(((-13.6 / (b * b)) - (-13.6 / (a * a))));
}
int main(void) {
const double expected = 1.8888888888888888;
const double actual = hydrogen_transition(3, 2);
assert(fabs(actual - expected) < 1e-6 * fmax(1.0, fabs(expected)));
}
C++
#include <cassert>
#include <cmath>
#include <numbers>
double hydrogen_transition(double a, double b) {
return std::fabs(((-13.6 / (b * b)) - (-13.6 / (a * a))));
}
int main() {
constexpr double expected = 1.8888888888888888;
const double actual = hydrogen_transition(3, 2);
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 hydrogen_transition(double a, double b)
; Linux x86-64 NASM · System V ABI · first eight doubles in xmm0–xmm7
extern fabs
global hydrogen_transition
section .text
hydrogen_transition:
push rbp
mov rbp, rsp
sub rsp, 80
movsd [rbp-8], xmm0
movsd [rbp-16], xmm1
mov rax, 0xc02b333333333333
movq xmm0, rax
movsd [rbp-48], xmm0
movsd xmm0, [rbp-16]
mulsd xmm0, [rbp-16]
movsd [rbp-56], xmm0
movsd xmm0, [rbp-48]
divsd xmm0, [rbp-56]
movsd [rbp-40], xmm0
mov rax, 0xc02b333333333333
movq xmm0, rax
movsd [rbp-72], xmm0
movsd xmm0, [rbp-8]
mulsd xmm0, [rbp-8]
movsd [rbp-80], xmm0
movsd xmm0, [rbp-72]
divsd xmm0, [rbp-80]
movsd [rbp-64], xmm0
movsd xmm0, [rbp-40]
subsd xmm0, [rbp-64]
movsd [rbp-32], xmm0
movsd xmm0, [rbp-32]
call fabs wrt ..plt
movsd [rbp-24], xmm0
movsd xmm0, [rbp-24]
leave
ret
MATLAB
function result = hydrogen_transition(a, b)
result = abs(((-13.6 / (b * b)) - (-13.6 / (a * a))));
end
Wolfram Language
ClearAll[mwCalculate];
mwCalculate[a_, b_] := Abs[((-13.6 / (b * b)) - (-13.6 / (a * a)))];
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). Hydrogen Energy Transition Calculator. MW SysArc Tools. https://math.mwsysarc.com/quantum-mathematics/hydrogen-energy-transition
MLA 9
MW SysArc. “Hydrogen Energy Transition Calculator.” MW SysArc Tools, 21 July 2026, https://math.mwsysarc.com/quantum-mathematics/hydrogen-energy-transition. Accessed 31 Aug. 2026.
Chicago 17
MW SysArc. “Hydrogen Energy Transition Calculator.” MW SysArc Tools. Published July 21, 2026. Accessed August 31, 2026. https://math.mwsysarc.com/quantum-mathematics/hydrogen-energy-transition.
Harvard
MW SysArc (2026) ‘Hydrogen Energy Transition Calculator’, MW SysArc Tools. Published 21 July 2026. Available at: https://math.mwsysarc.com/quantum-mathematics/hydrogen-energy-transition (Accessed: 31 August 2026).
BibTeX and RIS records
BibTeX
@misc{mwsysarc_hydrogen_transition_2026,
author = {{MW SysArc}},
title = {Hydrogen Energy Transition Calculator},
howpublished = {MW SysArc Tools},
year = {2026},
url = {https://math.mwsysarc.com/quantum-mathematics/hydrogen-energy-transition},
note = {Published July 21, 2026; accessed August 31, 2026}
}RIS
TY - ELEC
AU - MW SysArc
TI - Hydrogen Energy Transition Calculator
T2 - MW SysArc Tools
PY - 2026
DA - 2026-07-21
Y2 - 2026-08-31
UR - https://math.mwsysarc.com/quantum-mathematics/hydrogen-energy-transition
N1 - Published July 21, 2026
ER -Clear answers
Frequently asked questions
What does the Hydrogen transition do?
Calculate hydrogen energy levels and the photon energy and wavelength between two principal quantum numbers.
How does the Hydrogen transition work?
The calculator applies Eₙ=−13.6 eV/n²; |ΔE|=hc/λ. Hydrogen's allowed bound-state energies are discrete, so transitions emit or absorb photons with exactly the energy difference between levels.
What can I learn from the Hydrogen transition?
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 .