Normalized solution (mathematics)

<p>In mathematics, a normalized solution to an <a href="/facts/Ordinary_differential_equation/ygKGQ8kD">ordinary</a> or <a href="/facts/Partial_differential_equation/DyPsBlv7">partial differential equation</a> is a solution with prescribed norm, that is, a solution which satisfies a condition like 
  
    
      
        
          ∫
          
            
              
                R
              
              
                N
              
            
          
        
        
          |
        
        u
        (
        x
        )
        
          
            |
          
          
            2
          
        
        
        d
        x
        =
        1.
      
    
    {\displaystyle \int _{\mathbb {R} ^{N}}|u(x)|^{2}\,dx=1.}
  
 In this article, the normalized solution is introduced by using the <a href="/facts/Nonlinear_Schr%25C3%25B6dinger_equation/gfGkRApW">nonlinear Schrödinger equation</a>. The nonlinear <a href="/facts/Schr%25C3%25B6dinger_equation/3csgSIWF">Schrödinger equation</a> (NLSE) is a fundamental equation in <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> and other various fields of physics, describing the evolution of complex <a href="/facts/Wave_functions/KgM5ykjY">wave functions</a>. In Quantum Physics, normalization means that the total probability of finding a quantum particle anywhere in the universe is unity.
</p>

Normalized solution (mathematics) open-in-new

Normalized solution (mathematics)