How to calculate Bohr radius hydrogen-like atoms excitons

Calculate Bohr radius a_n = a_0*n^2/Z hydrogen-like atoms + Wannier-Mott exciton GaAs Si semiconductor. a_0 = 5.29e-11 m H 1s. Niels Bohr Denmark 1885-1962 Nobel Prize 1922 + 1913. NIST + JILA + Princeton + Caltech.

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How to calculate the Bohr radius (a_n) fundamental scale atomic physics and semiconductor physics measuring most probable orbital radius in hydrogen atom or hydrogen-like systems (single-electron atoms like He+ Li2+ Be3+) and Wannier-Mott excitons in semiconductors: a_n = (4*pi*epsilon_0*hbar^2)/(m*Z*e^2) * n^2 = a_0 * n^2 / Z.

Fundamental Bohr radius: a_0 = 5.29177e-11 m = 0.529 A = 0.0529 nm for H 1s ground state. n principal quantum number (1 ground; 2 3... excited Rydberg states); Z nuclear atomic number (H 1; He+ 2; Li2+ 3); m electron effective mass (m_e = 9.109e-31 kg free atoms; m_eff < m_e in semiconductor); epsilon_0 vacuum permittivity + epsilon_r effective in dielectric medium.

Wannier-Mott exciton semiconductor: a_eff = a_0 * (m_e/m_eff) * epsilon_r + typical GaAs a_eff = 12.9 nm with m_eff=0.067 + epsilon_r=12.9; Si a_eff = 4.5 nm with m_eff=0.26 + epsilon_r=11.9; CdSe a_eff = 5.6 nm; classical reference + Wannier 1937 + Mott 1938.

Rydberg binding energy Ry: E_n = -Ry * Z^2/n^2 = -13.6 eV * Z^2/n^2 for H atom + ground state -13.6 eV + Rydberg constant R_inf = 1.097e7 m^-1; orbital velocity: v_n = (Z/n) * alpha*c = (Z/n) * 2.19e6 m/s + alpha = 1/137 fine structure constant.

Bohr atomic model 1913: angular momentum quantization L = n*hbar + circular orbits + stationary states + photon emission E_photon = E_n - E_m + hydrogen spectroscopy Balmer Lyman Paschen series + classical reference.

Atomic units: a_0 atomic length unit; Hartree E_h = 2*Ry = 27.2 eV energy unit; tau_au = hbar/E_h time; fine structure alpha = e^2/(4*pi*epsilon_0*hbar*c) = 1/137.036.

Quantum mechanics extensions: Schrodinger 1926 wave equation orbitals 1s 2s 2p 3s... + radial probability density |psi|^2 + max at Bohr radius for H 1s; Heisenberg uncertainty 1927 + delta_x*delta_p >= hbar/2 + classical reference + atomic stability quantum.

Related numbers: Rydberg constant R_inf; fine structure alpha; Compton wavelength lambda_C = h/(m*c) = 2.43e-12 m electron; classical electron radius r_e = e^2/(4*pi*epsilon_0*m_e*c^2) = 2.82e-15 m; Hartree energy E_h; de Broglie wavelength lambda = h/p.

Quantum dot exciton in nanoparticles: quantum confinement Brus 1984 + R*sphere radius < a_eff Bohr exciton + classical reference + CdSe quantum dots colloidal Bawendi MIT 1990s + LED display applications Sharp Nanosys QD-OLED Samsung. History: Niels Henrik David Bohr (Copenhagen Denmark 1885 - Copenhagen 1962) Univ Copenhagen 1916-1962 + Niels Bohr Institute Copenhagen 1921 founder + Manchester with Rutherford 1912; 1913 paper Phil Mag 26:1 'On the constitution of atoms and molecules' + 1913 Phil Mag 26:476 + 1913 Phil Mag 26:857 classic Bohr atomic model + angular momentum quantization L = n*hbar + classical reference + Nobel Prize Physics 1922 'for his services in the investigation of the structure of atoms and of the radiation emanating from them' + classical reference + Niels Bohr Institute world-leading + Member US National Academy + Royal Society 1926 + Copley Medal 1938 + Atoms for Peace Award 1957 + Manhattan Project 1943-1945 + Bohr-Einstein debates EPR 1935.

Erwin Schrodinger (Vienna 1887-1961) Vienna + Berlin + Oxford + Dublin DIAS + Vienna 1956-1961; 1926 paper Annalen 79:361 'Quantisierung als Eigenwertproblem' Schrodinger equation + 1944 'What is Life?' Cambridge classical biology DNA + Nobel Prize Physics 1933 shared with Dirac + Schrodinger cat thought experiment 1935.

Werner Heisenberg (Wurzburg 1901 - Munich 1976) Univ Leipzig 1927-1942 + MPI Gottingen 1942-1970; 1925 Z Phys 33:879 matrix mechanics + 1927 Z Phys 43:172 classic Heisenberg uncertainty principle delta_x*delta_p >= hbar/2 + Nobel Prize Physics 1932 + Manhattan Project Germany Uranium Club.

Gregory Wannier (Switzerland-USA 1911-1983) Bell Labs + Univ Oregon; 1937 Phys Rev 52:191 classic Wannier exciton + Wannier function eponym.

Nevill Mott (UK 1905-1996) Cambridge + Bristol 1933-1954 + Cambridge Cavendish director 1954-1971; 1938 Trans Faraday Soc 34:500 Mott exciton + 1949 Mott transition Mott insulator + Nobel Prize Physics 1977 shared with Anderson Van Vleck + Royal Society + Copley Medal 1972. Legacy: Sakurai-Napolitano 'Modern Quantum Mechanics' 3rd 2020 Cambridge 568 pages classical + Griffiths-Schroeter 'Introduction to Quantum Mechanics' 3rd 2018 Cambridge 506 pages classical + Bransden-Joachain 'Physics of Atoms and Molecules' 2nd 2003 Pearson 1114 pages + Demtroder 'Atoms Molecules and Photons' 4th 2018 Springer 632 pages + Foot 'Atomic Physics' 2005 OUP 332 pages classical + Yu-Cardona 'Fundamentals of Semiconductors' 4th 2010 Springer 778 pages excitons + Klingshirn 'Semiconductor Optics' 4th 2012 Springer 802 pages + Klimov 'Optical Properties of Quantum Dots' 2010 Wiley 538 pages quantum dots + Reimann-Manninen 2002 Rev Mod Phys 74:1283.

USA research atomic physics + excitons + quantum dots: NIST Gaithersburg Boulder 4800 + atomic clocks Cs Rb world-leading + Rydberg constant precision SI 2019 redefinition; JILA Joint Institute Laboratory Astrophysics Boulder 600 + ultracold atoms Eric Cornell Carl Wieman Jun Ye + BEC 1995 Nobel Prize 2001 world-leading; NSF Materials 12000; DOE national labs Argonne ORNL LBNL LANL Sandia 35000; Sandia 13000 + atomic physics; LLNL 7500; LANL 13000 + quantum information; NASA Goddard JPL 17000 + atomic clocks navigation; NIH NIGMS 2500; NRL 2700.

USA universities atomic physics + excitons: MIT 11500 + Wolfgang Ketterle BEC Nobel 2001 world-leading + Vladan Vuletic + Mildred Dresselhaus legacy; Caltech 8800 + Linus Pauling alpha helix legacy + Hideo Mabuchi quantum optics; Stanford 11000 + Steven Chu cold atoms Nobel 1997 + Mark Kasevich atom interferometry; UC Berkeley 8500 + Daniel Kleppner laser cooling; Princeton 5400 + Jason Petta quantum dots; Harvard 22000 + John Doyle ultracold molecules + Mikhail Lukin quantum world-leading; Yale 13000; Cornell 5200; Univ Chicago 9000; Univ Colorado Boulder 35000 + JILA Eric Cornell Jun Ye Carl Wieman world-leading; Univ Wisconsin Madison 7000; Penn State 8500; Northwestern 7100.

USA industry semiconductor optoelectronics: Intel Santa Clara 132000; TSMC USA 8000; GlobalFoundries 13000; Texas Instruments 33000; Applied Materials 35000; Lam Research 19000; Coherent II-VI Corning 22000 + lasers semiconductor; IPG Photonics 4500 + fiber lasers world-leading; Lumentum San Jose CA 6500 + telecom laser.

USA industry quantum dots LED display: Nanosys Milpitas CA 100 + CdSe quantum dots LED display world-leading; QD Vision Lexington MA 100 + Samsung QD-OLED display; Quantum Materials Corporation San Marcos TX 50; 3M 92000 + display films.

USA industry atomic clocks + GPS: Microsemi Aliso Viejo CA 4500 + atomic clocks GPS world-leading; Symmetricom Microsemi 1000; Microchip Technology 22000; Lockheed Martin 122000 + GPS III; L3Harris Melbourne 47000; Northrop Grumman 95000; NIST 4800.

USA industry quantum computing trapped ions: IonQ College Park MD 250 + trapped ion quantum computer world-leading; Honeywell Quantum Solutions 100 + Quantinuum; Atom Computing Berkeley CA 200; QuEra Computing Boston 200 + neutral atoms; Pasqal France USA.

Bohr formula

Definition: a_n = (4*pi*epsilon_0*hbar^2)/(m*Z*e^2) * n^2 = a_0 * n^2 / Z. a_0 = 5.29177e-11 m fundamental Bohr radius H 1s + n principal + Z atomic.

Wannier-Mott effective exciton: a_eff = a_0 * (m_e/m_eff) * epsilon_r. GaAs a_eff = 12.9 nm; Si 4.5 nm; CdSe 5.6 nm.

Rydberg binding energy: E_n = -Ry * Z^2/n^2. Ry = 13.6 eV + H ground state -13.6 eV + He+ -54.4 eV.

Orbital velocity: v_n = (Z/n) * alpha*c = (Z/n) * 2.19e6 m/s + alpha = 1/137.

Typical values: H 1s a_0 = 0.529 A; H 2s 2.12 A; H 3s 4.76 A Rydberg; He+ 1s 0.265 A; Li2+ 1s 0.176 A; GaAs exciton 12.9 nm; Si exciton 4.5 nm; CdSe quantum dot 5.6 nm.

History and context

Niels Henrik David Bohr (Copenhagen 1885-1962) Univ Copenhagen + Niels Bohr Institute 1921 founder + Manchester 1912 with Rutherford; 1913 paper Phil Mag 26:1 'On the constitution of atoms and molecules' classic Bohr atomic model + L = n*hbar quantization + classical reference + Nobel Prize Physics 1922 + Niels Bohr Institute world-leading + Member US National Academy + Royal Society 1926 + Copley Medal 1938 + Atoms for Peace 1957 + Manhattan Project 1943-1945 + Bohr-Einstein debates EPR 1935.

Erwin Schrodinger (1887-1961); 1926 Annalen 79:361 Schrodinger equation + 1944 'What is Life?' + Nobel Prize Physics 1933 shared with Dirac + Schrodinger cat 1935.

Werner Heisenberg (1901-1976) Leipzig + MPI Gottingen; 1925 matrix mechanics + 1927 uncertainty principle + Nobel Prize Physics 1932.

Gregory Wannier (1911-1983) Bell Labs + Oregon; 1937 Phys Rev 52:191 classic Wannier exciton + Wannier function eponym.

Nevill Mott (UK 1905-1996) Bristol + Cambridge Cavendish director 1954-1971; 1938 Trans Faraday Soc + 1949 Mott transition + Nobel Prize Physics 1977 + Royal Society + Copley Medal 1972. Legacy: Sakurai-Napolitano 2020 Cambridge 568 + Griffiths-Schroeter 2018 Cambridge 506 + Bransden-Joachain 2003 Pearson 1114 + Demtroder 2018 Springer 632 + Foot 2005 OUP 332 + Yu-Cardona 2010 Springer 778 excitons + Klingshirn 2012 Springer 802 + Klimov 2010 Wiley 538 quantum dots.

USA research: NIST 4800 + atomic clocks Cs Rb world-leading + R_inf SI 2019; JILA Boulder 600 + Cornell Wieman Ye BEC Nobel 2001 world-leading; NSF Materials 12000; DOE labs 35000; Sandia 13000; LLNL 7500; LANL 13000; NASA 17000; NRL 2700.

Universities: MIT 11500 + Ketterle BEC Nobel 2001 world-leading + Vuletic; Caltech 8800 + Pauling legacy + Mabuchi; Stanford 11000 + Chu Nobel 1997 cold atoms + Kasevich; Harvard 22000 + Doyle + Lukin world-leading; Yale 13000; Cornell 5200; Princeton 5400 + Petta quantum dots; Chicago 9000; Colorado Boulder 35000 + JILA Cornell Wieman Ye world-leading; Wisconsin 7000; Penn State 8500; Northwestern 7100.

Industry semiconductor: Intel 132000; TSMC USA 8000; GlobalFoundries 13000; Texas Instruments 33000; Applied Materials 35000; Lam Research 19000; Coherent II-VI 22000; IPG Photonics 4500 world-leading; Lumentum 6500.

QD LED: Nanosys 100 world-leading + Samsung QD-OLED; Quantum Materials 50; 3M 92000.

Atomic clocks GPS: Microsemi 4500 world-leading; Microchip 22000; Lockheed Martin 122000 GPS III; L3Harris 47000; NIST 4800.

Quantum computing: IonQ 250 world-leading trapped ion; Honeywell Quantinuum 100; Atom Computing 200; QuEra 200 neutral atoms.

Bohr values table

  • a_n = a_0*n^2/Z: definition
  • a_0 = 5.29e-11 m: H 1s
  • E_n = -Ry*Z^2/n^2: Rydberg
  • v_n = (Z/n)*alpha*c: velocity
  • alpha = 1/137: fine structure
  • Exciton: a_eff = a_0*(m/m_eff)*er
  • H 1s: a_0 = 0.529 A
  • He+: 0.265 A
  • GaAs exciton: 12.9 nm
  • Si exciton: 4.5 nm

5 numerical examples

  1. Hydrogen H 1s ground state: n=1 + Z=1 + a_n = 0.529 A + E_n = -13.6 eV + classical reference Bohr 1913.
  2. Helium He+ 1s: n=1 + Z=2 + a_n = 0.265 A + E_n = -54.4 eV + tightly bound.
  3. Hydrogen H n=2 Rydberg: n=2 + Z=1 + a_n = 2.12 A + E_n = -3.4 eV + excited state Balmer series.
  4. GaAs Wannier-Mott exciton: n=1 + Z=1 + m_eff=0.067 + er=12.9 + a_eff = 12.9 nm + E_b = -5 meV classical reference HEMT laser optoelectronics.
  5. Silicon Si exciton indirect bandgap: n=1 + Z=1 + m_eff=0.26 + er=11.9 + a_eff = 4.5 nm + E_b = -25 meV classical reference photovoltaic.

Limitations

Semiclassical Bohr model: circular orbits + quantization + correct only for hydrogen-like single-electron atoms + not for multi-electron atoms + Sakurai 2020.

Schrodinger 1926 correct: orbital wavefunctions + radial probability density + max at Bohr radius for H 1s + Griffiths 2018.

Relativistic corrections: fine structure + spin-orbit + Lamb shift + Bransden-Joachain 2003.

Multi-electron atoms: Hartree-Fock + DFT + Z_eff effective + Demtroder 2018.

Quantum confinement: nanoparticles R < a_eff Brus 1984 + Klimov 2010.

Common mistakes

  • Bohr semiclassical vs Schrodinger quantum: Bohr circular hydrogen-like only + Schrodinger orbital wavefunction full quantum + Sakurai 2020 distinct.
  • Multi-electron atoms: a_0 valid only H He+ Li2+ Be3+ + multi atoms not correct + Z_eff effective + classical distinction.
  • Exciton vs free atom: exciton in dielectric medium modified a_eff = a_0*(m_e/m_eff)*er + Wannier-Mott classical reference.
  • Atomic units vs SI: a_0 = 5.29e-11 m SI vs 1 in atomic units + classical distinction.
  • Binding energy sign: E_n = -Ry*Z^2/n^2 negative bound state + classical distinction.

Applications

  1. Hydrogen-like atom spectroscopy: hydrogen H + ionized helium He+ + Lyman Balmer Paschen series + classical reference.
  2. GaAs HEMT semiconductor excitons: Intel + GaN + optoelectronics laser LED.
  3. CdSe colloidal quantum dots: Nanosys world-leading + LED display Samsung QD-OLED.
  4. Si photovoltaic exciton: First Solar + Tesla Solar + thin film silicon photovoltaic + a_eff Si 4.5 nm.
  5. Atomic clocks Cs Rb: NIST world-leading + Microsemi + GPS III time standard SI 2019.
  6. Cold atoms quantum sensors: JILA Boulder + atom interferometry gravity + Stanford Kasevich.
  7. Quantum computing trapped ions: IonQ world-leading + Honeywell Quantinuum + neutral atoms QuEra.
  8. Atomic physics fundamental constants: NIST + Rydberg constant R_inf precision metrology SI 2019.
  9. Astronomical spectroscopy: Lyman alpha + classical reference + NASA Hubble JWST.
  10. Bose-Einstein condensation BEC: ultracold atoms + JILA Cornell Wieman Nobel 2001 + Ketterle MIT Nobel 2001.

Updated July 2026

Frequently Asked Questions

What is the Bohr radius a_0?

Bohr radius a_0 = 5.29177e-11 m = 0.529 A = 0.0529 nm defines fundamental atomic physics scale. Niels Bohr 1913 Phil Mag 26:1 + Nobel Prize Physics 1922. For hydrogen-like atoms a_n = a_0*n^2/Z with n principal quantum Z atomic; binding energy E_n = -13.6 eV*Z^2/n^2 Rydberg. For Wannier-Mott excitons in semiconductors a_eff = a_0*(m_e/m_eff)*er + GaAs 12.9 nm + Si 4.5 nm + CdSe 5.6 nm classical reference.

Difference between Bohr semiclassical and Schrodinger atom?

Bohr 1913 semiclassical model: circular quantized orbits L=n*hbar + correct only for single-electron hydrogen-like atoms H He+ Li2+ Be3+. Schrodinger 1926 quantum model: wave equation + orbital wavefunctions psi(r) + radial probability density |psi|^2 peak at Bohr radius for H 1s + correct for multi-electron atoms via Hartree-Fock + DFT. Schrodinger Nobel Prize 1933 + classical Sakurai-Napolitano 2020 Cambridge 568 pages.

What is a Wannier-Mott exciton?

Wannier-Mott exciton bound electron-hole pair in semiconductor with large dielectric constant + small effective mass + a_eff >> a_0. Typical GaAs a_eff = 12.9 nm with m_eff=0.067 m_e + er=12.9 + E_b = -5 meV; Si a_eff = 4.5 nm + E_b = -25 meV; CdSe a_eff = 5.6 nm. Wannier 1937 Phys Rev 52:191 + Mott 1938 Trans Faraday Soc 34:500 + classical reference Yu-Cardona 2010 Springer 778 pages.

What is angular momentum quantization?

Angular momentum quantization L = n*hbar postulated by Niels Bohr 1913 in his atomic model. Explains atomic stationary states and discrete spectral lines of hydrogen Lyman Balmer Paschen series. Niels Bohr Nobel Prize Physics 1922 + L = n*hbar with n=1,2,3... + classical reference + correctly derived later from Schrodinger 1926 wave equation with angular quantum number l + classical reference Bransden-Joachain 2003.

What is the fine structure constant alpha?

Fine structure constant alpha = e^2/(4*pi*epsilon_0*hbar*c) = 1/137.036 measures dimensionless electromagnetic interaction strength. Typical value 1/137 measured 1e-12 precision + NIST + JILA Boulder + classical reference SI 2019 redefinition. Bohr orbital velocity v_n = (Z/n)*alpha*c = (Z/n)*0.0073*c. Sommerfeld 1916 fine-structure relativistic correction + classical reference Foot 2005 OUP 332 pages atomic physics.

What are CdSe quantum dots and applications?

CdSe quantum dots colloidal nanoparticles 2-10 nm with quantum confinement R_sphere < a_eff Wannier-Mott exciton. Brus 1984 effect + Bawendi MIT 1990s synthesis + Nobel Prize 2023 Bawendi Brus Ekimov + applications: LED display Nanosys QD-OLED Samsung; biological imaging non-toxic coating; solar cells. Nanosys world-leading + classical reference Klimov 2010 Wiley 538 pages quantum dots.

Bohr radius industrial applications?

Applications: hydrogen-like atom spectroscopy Lyman Balmer Paschen series classical reference; GaAs HEMT semiconductor excitons Intel + GaN + optoelectronics; CdSe colloidal quantum dots Nanosys world-leading + Samsung QD-OLED; Si photovoltaic exciton First Solar + Tesla Solar; atomic clocks Cs Rb NIST world-leading + GPS III SI 2019; cold atoms quantum sensors JILA + Stanford Kasevich; quantum computing trapped ions IonQ world-leading + Honeywell Quantinuum + neutral atoms QuEra.

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