Separable partial differential equation

<h2 id="example">Example</h2>
<p>For example, consider the time-independent <a href="/facts/Schr%C3%B6dinger_equation/3csgSIWF">Schrödinger equation</a>
</p>

[
        −
        
          ∇
          
            2
          
        
        +
        V
        (
        
          x
        
        )
        ]
        ψ
        (
        
          x
        
        )
        =
        E
        ψ
        (
        
          x
        
        )
      
    
    {\displaystyle [-\nabla ^{2}+V(\mathbf {x} )]\psi (\mathbf {x} )=E\psi (\mathbf {x} )}

<p>for the function 
  
    
      
        ψ
        (
        
          x
        
        )
      
    
    {\displaystyle \psi (\mathbf {x} )}
  
 (in dimensionless units, for simplicity).  (Equivalently, consider the inhomogeneous <a href="/facts/Helmholtz_equation/SWfaXrTB">Helmholtz equation</a>.)  If the function 
  
    
      
        V
        (
        
          x
        
        )
      
    
    {\displaystyle V(\mathbf {x} )}
  
 in three dimensions is of the form
</p>

V
        (
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ,
        
          x
          
            3
          
        
        )
        =
        
          V
          
            1
          
        
        (
        
          x
          
            1
          
        
        )
        +
        
          V
          
            2
          
        
        (
        
          x
          
            2
          
        
        )
        +
        
          V
          
            3
          
        
        (
        
          x
          
            3
          
        
        )
        ,
      
    
    {\displaystyle V(x_{1},x_{2},x_{3})=V_{1}(x_{1})+V_{2}(x_{2})+V_{3}(x_{3}),}

<p>then it turns out that the problem can be separated into three one-dimensional ODEs for functions 
  
    
      
        
          ψ
          
            1
          
        
        (
        
          x
          
            1
          
        
        )
      
    
    {\displaystyle \psi _{1}(x_{1})}
  
, 
  
    
      
        
          ψ
          
            2
          
        
        (
        
          x
          
            2
          
        
        )
      
    
    {\displaystyle \psi _{2}(x_{2})}
  
, and 
  
    
      
        
          ψ
          
            3
          
        
        (
        
          x
          
            3
          
        
        )
      
    
    {\displaystyle \psi _{3}(x_{3})}
  
, and the final solution can be written as 
  
    
      
        ψ
        (
        
          x
        
        )
        =
        
          ψ
          
            1
          
        
        (
        
          x
          
            1
          
        
        )
        ⋅
        
          ψ
          
            2
          
        
        (
        
          x
          
            2
          
        
        )
        ⋅
        
          ψ
          
            3
          
        
        (
        
          x
          
            3
          
        
        )
      
    
    {\displaystyle \psi (\mathbf {x} )=\psi _{1}(x_{1})\cdot \psi _{2}(x_{2})\cdot \psi _{3}(x_{3})}
  
.  (More generally, the separable cases of the Schrödinger equation were enumerated by Eisenhart in 1948.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>)
</p>

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

<ol>
<li id="fn:1"><p>Eisenhart, L. P. (1948-07-01). "Enumeration of Potentials for Which One-Particle Schroedinger Equations Are Separable". Physical Review. 74 (1). American Physical Society (APS): 87–89. Bibcode:1948PhRv...74...87E. doi:10.1103/physrev.74.87. ISSN 0031-899X. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Separable partial differential equation open-in-new

Separable partial differential equation