Magnetic quantum number

<h2 id="derivation">Derivation</h2>

<p>There is a set of quantum numbers associated with the energy states of the atom. The four quantum numbers 
  
    
      
        n
      
    
    {\displaystyle n}
  
, 
  
    
      
        ℓ
      
    
    {\displaystyle \ell }
  
, 
  
    
      
        
          m
          
            l
          
        
      
    
    {\displaystyle m_{l}}
  
, and 
  
    
      
        
          m
          
            s
          
        
      
    
    {\displaystyle m_{s}}
  
 specify the complete <a href="/facts/Quantum_state/7jbxd7Dx">quantum state</a> of a single electron in an atom called its <a href="/facts/Wavefunction/KgM5ykjY">wavefunction</a> or orbital.  The <a href="/facts/Schr%25C3%25B6dinger_equation/3csgSIWF">Schrödinger equation</a> for the wavefunction of an atom with one electron is a <a href="/facts/Separable_partial_differential_equation/grsuzbv2">separable partial differential equation</a>.  (This is not the case for the neutral <a href="/facts/Helium_atom/ASrTuP3v">helium atom</a> or other atoms with mutually interacting electrons, which require more sophisticated methods for solution<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>)  This means that the wavefunction as expressed in <a href="/facts/Spherical_coordinates/5cVnNof3">spherical coordinates</a> can be broken down into the product of three functions of the radius, colatitude (or polar) angle, and azimuth:<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>
</p>

ψ
        (
        r
        ,
        θ
        ,
        ϕ
        )
        =
        R
        (
        r
        )
        P
        (
        θ
        )
        F
        (
        ϕ
        )
      
    
    {\displaystyle \psi (r,\theta ,\phi )=R(r)P(\theta )F(\phi )}

<p>The differential equation for 
  
    
      
        F
      
    
    {\displaystyle F}
  
 can be solved in the form 
  
    
      
        F
        (
        ϕ
        )
        =
        A
        
          e
          
            λ
            ϕ
          
        
      
    
    {\displaystyle F(\phi )=Ae^{\lambda \phi }}
  
.  Because values of the azimuth angle 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
  
 differing by 2
  
    
      
        π
      
    
    {\displaystyle \pi }
  
 <a href="/facts/Radians/Srd38Sxa">radians</a> (360 degrees) represent the same position in space, and the overall magnitude of 
  
    
      
        F
      
    
    {\displaystyle F}
  
 does not grow with arbitrarily large 
  
    
      
        ϕ
      
    
    {\displaystyle \phi }
  
 as it would for a real exponent, the coefficient 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 must be quantized to integer multiples of 
  
    
      
        i
      
    
    {\displaystyle i}
  
, producing an <a href="/facts/Imaginary_exponent/pac6QY2g">imaginary exponent</a>: 
  
    
      
        λ
        =
        i
        
          m
          
            l
          
        
      
    
    {\displaystyle \lambda =im_{l}}
  
.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>  These integers are the magnetic quantum numbers. The same constant appears in the colatitude equation, where larger values of 
  
    
      
        
          
            
              m
              
                l
              
            
          
          
            2
          
        
      
    
    {\displaystyle {m_{l}}^{2}}
  
 tend to decrease the magnitude of 
  
    
      
        P
        (
        θ
        )
        ,
      
    
    {\displaystyle P(\theta ),}
  
 and values of 
  
    
      
        
          m
          
            l
          
        
      
    
    {\displaystyle m_{l}}
  
 greater than the azimuthal quantum number 
  
    
      
        ℓ
      
    
    {\displaystyle \ell }
  
 do not permit any solution for 
  
    
      
        P
        (
        θ
        )
        .
      
    
    {\displaystyle P(\theta ).}

</p>
<table><tbody><tr><th colspan="4">Relationship between Quantum Numbers</th></tr><tr><th>Orbital</th><th>Values</th><th>Number of Values for                               m                      l                                {\displaystyle m_{l}}  <a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a></th><th>Electrons per subshell</th></tr><tr><th>s</th><td>                    ℓ        =        0        ,                          m                      l                          =        0              {\displaystyle \ell =0,\quad m_{l}=0}  </td><td>1</td><td>2</td></tr><tr><th>p</th><td>                    ℓ        =        1        ,                          m                      l                          =        −        1        ,        0        ,        +        1              {\displaystyle \ell =1,\quad m_{l}=-1,0,+1}  </td><td>3</td><td>6</td></tr><tr><th>d</th><td>                    ℓ        =        2        ,                          m                      l                          =        −        2        ,        −        1        ,        0        ,        +        1        ,        +        2              {\displaystyle \ell =2,\quad m_{l}=-2,-1,0,+1,+2}  </td><td>5</td><td>10</td></tr><tr><th>f</th><td>                    ℓ        =        3        ,                          m                      l                          =        −        3        ,        −        2        ,        −        1        ,        0        ,        +        1        ,        +        2        ,        +        3              {\displaystyle \ell =3,\quad m_{l}=-3,-2,-1,0,+1,+2,+3}  </td><td>7</td><td>14</td></tr><tr><th>g</th><td>                    ℓ        =        4        ,                          m                      l                          =        −        4        ,        −        3        ,        −        2        ,        −        1        ,        0        ,        +        1        ,        +        2        ,        +        3        ,        +        4              {\displaystyle \ell =4,\quad m_{l}=-4,-3,-2,-1,0,+1,+2,+3,+4}  </td><td>9</td><td>18</td></tr></tbody></table>
<h2 id="as-a-component-of-angular-momentum">As a component of angular momentum</h2>

<p>The axis used for the polar coordinates in this analysis is chosen arbitrarily. The quantum number 
  
    
      
        
          m
          
            l
          
        
      
    
    {\displaystyle m_{l}}
  
 refers to the projection of the angular momentum in this arbitrarily-chosen direction, conventionally called the 
  
    
      
        z
      
    
    {\displaystyle z}
  
-direction or quantization axis. 
  
    
      
        
          L
          
            z
          
        
      
    
    {\displaystyle L_{z}}
  
, the magnitude of the angular momentum in the 
  
    
      
        z
      
    
    {\displaystyle z}
  
-direction, is given by the formula:<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>
</p>

L
          
            z
          
        
        =
        
          m
          
            l
          
        
        ℏ
      
    
    {\displaystyle L_{z}=m_{l}\hbar }
  
.
<p>This is a component of the atomic electron's total orbital angular momentum 
  
    
      
        
          L
        
      
    
    {\displaystyle \mathbf {L} }
  
, whose magnitude is related to the azimuthal quantum number of its subshell 
  
    
      
        ℓ
      
    
    {\displaystyle \ell }
  
 by the equation:
</p>

L
        =
        ℏ
        
          
            ℓ
            (
            ℓ
            +
            1
            )
          
        
      
    
    {\displaystyle L=\hbar {\sqrt {\ell (\ell +1)}}}
  
,
<p>where 
  
    
      
        ℏ
      
    
    {\displaystyle \hbar }
  
 is the <a href="/facts/Reduced_Planck_constant/qGHAe3qq">reduced Planck constant</a>. Note that this 
  
    
      
        L
        =
        0
      
    
    {\displaystyle L=0}
  
 for 
  
    
      
        ℓ
        =
        0
      
    
    {\displaystyle \ell =0}
  
 and approximates 
  
    
      
        L
        =
        
          (
          
            ℓ
            +
            
              
                
                  1
                  2
                
              
            
          
          )
        
        ℏ
      
    
    {\displaystyle L=\left(\ell +{\tfrac {1}{2}}\right)\hbar }
  
 for high 
  
    
      
        ℓ
      
    
    {\displaystyle \ell }
  
. It is not possible to measure the angular momentum of the electron along all three axes simultaneously. These properties were first demonstrated in the <a href="/facts/Stern%25E2%2580%2593Gerlach_experiment/PCWPE56v">Stern–Gerlach experiment</a>, by <a href="/facts/Otto_Stern/9eP2K5rB">Otto Stern</a> and <a href="/facts/Walther_Gerlach/qEjsbn0c">Walther Gerlach</a>.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a>
</p>
<h2 id="effect-in-magnetic-fields">Effect in magnetic fields</h2>
<p>The quantum number 
  
    
      
        
          m
          
            l
          
        
      
    
    {\displaystyle m_{l}}
  
 refers, loosely, to the direction of the  <a href="/facts/Angular_momentum/7TuSVFxq">angular momentum</a> <a href="/facts/Vector_(geometric)/bCLrdksy">vector</a>.  The magnetic quantum number 
  
    
      
        
          m
          
            l
          
        
      
    
    {\displaystyle m_{l}}
  
 only affects the electron's energy if it is in a magnetic field because in the absence of one, all spherical harmonics corresponding to the different arbitrary values of 
  
    
      
        
          m
          
            l
          
        
      
    
    {\displaystyle m_{l}}
  
 are equivalent.  The magnetic quantum number determines the energy shift of an <a href="/facts/Atomic_orbital/tKhgl6mL">atomic orbital</a> due to an external magnetic field (the <a href="/facts/Zeeman_effect/A6NMlp2D">Zeeman effect</a>) — hence the name <i>magnetic</i> quantum number.  However, the actual <a href="/facts/Magnetic_dipole_moment/VpAQKJtr">magnetic dipole moment</a> of an electron in an atomic orbital arises not only from the electron angular momentum but also from the electron spin, expressed in the spin quantum number.
</p><p>Since each electron has a magnetic moment in a magnetic field, it will be subject to a torque which tends to make the vector 
  
    
      
        
          L
        
      
    
    {\displaystyle \mathbf {L} }
  
 parallel to the field, a phenomenon known as <a href="/facts/Larmor_precession/KxkHsM76">Larmor precession</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Quantum_number/AByP6F0y">Quantum number</a>
<ul><li><a href="/facts/Azimuthal_quantum_number/vHSIzbFC">Azimuthal quantum number</a></li>
<li><a href="/facts/Principal_quantum_number/8SzSWV6w">Principal quantum number</a></li>
<li><a href="/facts/Spin_quantum_number/5Nh2H8gr">Spin quantum number</a></li>
<li><a href="/facts/Total_angular_momentum_quantum_number/onphLcJI">Total angular momentum quantum number</a></li></ul></li>
<li><a href="/facts/Electron_shell/QxI30eh9">Electron shell</a></li>
<li><a href="/facts/Basic_quantum_mechanics/p57bzH1Y">Basic quantum mechanics</a></li>
<li><a href="/facts/Bohr_atom/FmTHjaxy">Bohr atom</a></li>
<li><a href="/facts/Schr%25C3%25B6dinger_equation/3csgSIWF">Schrödinger equation</a></li></ul>
<h2 id="notes">Notes</h2>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>m is often used when only one kind of magnetic quantum number, such as ml or mj, is used in a text. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Martin, W. C.; Wiese, W. L. (2019). "Atomic Spectroscopy - A Compendium of Basic Ideas, Notation, Data, and Formulas". National Institute of Standards and Technology, Physical Measurement Laboratory. NIST. Retrieved 17 May 2023. <a href="https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas" target="_blank">https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Atkins, Peter William (1991). Quanta: A Handbook of Concepts (2nd ed.). Oxford University Press, USA. p. 297. ISBN 0-19-855572-5. <a href="0-19-855572-5" target="_blank">0-19-855572-5</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Atkins, Peter William (1991). Quanta: A Handbook of Concepts (2nd ed.). Oxford University Press, USA. p. 297. ISBN 0-19-855572-5. <a href="0-19-855572-5" target="_blank">0-19-855572-5</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Griffiths, David J. (2005). Introduction to quantum mechanics (2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall. pp. 136–137. ISBN 0-13-111892-7. OCLC 53926857. <a href="0-13-111892-7" target="_blank">0-13-111892-7</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Martin, W. C.; Wiese, W. L. (2019). "Atomic Spectroscopy - A Compendium of Basic Ideas, Notation, Data, and Formulas". National Institute of Standards and Technology, Physical Measurement Laboratory. NIST. Retrieved 17 May 2023. <a href="https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas" target="_blank">https://www.nist.gov/pml/atomic-spectroscopy-compendium-basic-ideas-notation-data-and-formulas</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Atkins, Peter William (1991). Quanta: A Handbook of Concepts (2nd ed.). Oxford University Press, USA. p. 297. ISBN 0-19-855572-5. <a href="0-19-855572-5" target="_blank">0-19-855572-5</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Atkins, Peter William (1991). Quanta: A Handbook of Concepts (2nd ed.). Oxford University Press, USA. p. 297. ISBN 0-19-855572-5. <a href="0-19-855572-5" target="_blank">0-19-855572-5</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>"Helium atom". 2010-07-20. <a href="http://farside.ph.utexas.edu/teaching/qmech/Quantum/node128.html" target="_blank">http://farside.ph.utexas.edu/teaching/qmech/Quantum/node128.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>"Hydrogen Schrodinger Equation". hyperphysics.phy-astr.gsu.edu. <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c3" target="_blank">http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydsch.html#c3</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>"Hydrogen Schrodinger Equation". hyperphysics.phy-astr.gsu.edu. <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydazi.html" target="_blank">http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydazi.html</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Herzberg, Gerhard (1950). Molecular Spectra and Molecular Structure (2 ed.). D van Nostrand Company. pp. 17–18. <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Herzberg, Gerhard (1950). Molecular Spectra and Molecular Structure (2 ed.). D van Nostrand Company. pp. 17–18. <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
<li id="fn:14"><p>"Spectroscopy: angular momentum quantum number". Encyclopædia Britannica. <a href="http://www.britannica.com/science/spectroscopy/Types-of-electromagnetic-radiation-sources#ref620216" target="_blank">http://www.britannica.com/science/spectroscopy/Types-of-electromagnetic-radiation-sources#ref620216</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li>
</ol>

Magnetic quantum number open-in-new

Magnetic quantum number