---
title: "Bohr–Sommerfeld model"
chunk: 2/2
source: "https://en.wikipedia.org/wiki/Bohr–Sommerfeld_model"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T16:28:10.422419+00:00"
instance: "kb-cron"
---

  
    
      
        
          
            
              
                d
                
                  2
                
              
              u
            
            
              d
              
                φ
                
                  2
                
              
            
          
        
        =
        −
        
          (
          
            1
            −
            
              k
              
                2
              
            
            
              
                
                  
                    Z
                    
                      2
                    
                  
                  
                    e
                    
                      4
                    
                  
                
                
                  
                    c
                    
                      2
                    
                  
                  
                    p
                    
                      
                        φ
                      
                    
                    
                      2
                    
                  
                
              
            
          
          )
        
        u
        +
        
          
            
              
                m
                
                  
                    0
                  
                
              
              k
              Z
              
                e
                
                  2
                
              
            
            
              p
              
                
                  φ
                
              
              
                2
              
            
          
        
        
          (
          
            1
            +
            
              
                W
                
                  
                    m
                    
                      
                        0
                      
                    
                  
                  
                    c
                    
                      2
                    
                  
                
              
            
          
          )
        
        =
        −
        
          ω
          
            
              0
            
          
          
            2
          
        
        u
        +
        K
      
    
    {\displaystyle {\frac {d^{2}u}{d\varphi ^{2}}}=-\left(1-k^{2}{\frac {Z^{2}e^{4}}{c^{2}p_{\mathrm {\varphi } }^{2}}}\right)u+{\frac {m_{\mathrm {0} }kZe^{2}}{p_{\mathrm {\varphi } }^{2}}}\left(1+{\frac {W}{m_{\mathrm {0} }c^{2}}}\right)=-\omega _{\mathrm {0} }^{2}u+K}
  

with solution

  
    
      
        u
        =
        
          
            1
            r
          
        
        =
        K
        +
        A
        cos
        ⁡
        
          ω
          
            
              0
            
          
        
        φ
      
    
    {\displaystyle u={\frac {1}{r}}=K+A\cos \omega _{\mathrm {0} }\varphi }
  

The angular shift of periapsis per revolution is given by

  
    
      
        
          φ
          
            
              s
            
          
        
        =
        2
        π
        
          (
          
            
              
                1
                
                  ω
                  
                    
                      0
                    
                  
                
              
            
            −
            1
          
          )
        
        ≈
        4
        
          π
          
            3
          
        
        
          k
          
            2
          
        
        
          
            
              
                Z
                
                  2
                
              
              
                e
                
                  4
                
              
            
            
              
                c
                
                  2
                
              
              
                n
                
                  
                    φ
                  
                
                
                  2
                
              
              
                h
                
                  2
                
              
            
          
        
      
    
    {\displaystyle \varphi _{\mathrm {s} }=2\pi \left({\frac {1}{\omega _{\mathrm {0} }}}-1\right)\approx 4\pi ^{3}k^{2}{\frac {Z^{2}e^{4}}{c^{2}n_{\mathrm {\varphi } }^{2}h^{2}}}}
  

With the quantum conditions

  
    
      
        ∮
        
          p
          
            
              φ
            
          
        
        
        d
        φ
        =
        2
        π
        
          p
          
            
              φ
            
          
        
        =
        
          n
          
            
              φ
            
          
        
        h
      
    
    {\displaystyle \oint p_{\mathrm {\varphi } }\,d\varphi =2\pi p_{\mathrm {\varphi } }=n_{\mathrm {\varphi } }h}
  

and

  
    
      
        ∮
        
          p
          
            
              r
            
          
        
        
        d
        r
        =
        
          p
          
            
              φ
            
          
        
        ∮
        
          
            (
            
              
                
                  1
                  r
                
              
              
                
                  
                    d
                    r
                  
                  
                    d
                    φ
                  
                
              
            
            )
          
          
            2
          
        
        
        d
        φ
        =
        
          n
          
            
              r
            
          
        
        h
      
    
    {\displaystyle \oint p_{\mathrm {r} }\,dr=p_{\mathrm {\varphi } }\oint \left({\frac {1}{r}}{\frac {dr}{d\varphi }}\right)^{2}\,d\varphi =n_{\mathrm {r} }h}
  

we will obtain energies

  
    
      
        
          
            W
            
              
                m
                
                  
                    0
                  
                
              
              
                c
                
                  2
                
              
            
          
        
        =
        
          
            (
            
              1
              +
              
                
                  
                    
                      α
                      
                        2
                      
                    
                    
                      Z
                      
                        2
                      
                    
                  
                  
                    
                      (
                      
                        
                          n
                          
                            
                              r
                            
                          
                        
                        +
                        
                          
                            
                              n
                              
                                
                                  φ
                                
                              
                              
                                2
                              
                            
                            −
                            
                              α
                              
                                2
                              
                            
                            
                              Z
                              
                                2
                              
                            
                          
                        
                      
                      )
                    
                    
                      2
                    
                  
                
              
            
            )
          
          
            −
            1
            
              /
            
            2
          
        
        −
        1
      
    
    {\displaystyle {\frac {W}{m_{\mathrm {0} }c^{2}}}=\left(1+{\frac {\alpha ^{2}Z^{2}}{\left(n_{\mathrm {r} }+{\sqrt {n_{\mathrm {\varphi } }^{2}-\alpha ^{2}Z^{2}}}\right)^{2}}}\right)^{-1/2}-1}
  

where 
  
    
      
        α
      
    
    {\displaystyle \alpha }
  
 is the fine-structure constant. This solution (using substitutions for quantum numbers) is equivalent to the solution of the Dirac equation. Nevertheless, both solutions fail to predict the Lamb shifts.

== See also ==

Bohr model
Old quantum theory

== References ==