Quantum LC circuit

An LC circuit can be quantized using the same methods as for the <a href="/facts/Harmonic_potential/JYA5J0h5">quantum harmonic oscillator</a>. An LC circuit is a variety of resonant circuit, and consists of an <a href="/facts/Inductor/YnfAXB3x">inductor</a>, represented by the letter L, and a <a href="/facts/Capacitor/hyowqv7n">capacitor</a>, represented by the letter C. When connected together, an <a href="/facts/Electric_current/Dc1xdnUX">electric current</a> can alternate between them at the circuit's <a href="/facts/Resonant_frequency/q7VLALUz">resonant frequency</a>:

ω
        =
        
          
            
              1
              
                L
                C
              
            
          
        
      
    
    {\displaystyle \omega ={\sqrt {1 \over LC}}}

where L is the <a href="/facts/Inductance/oIxj7JxX">inductance</a> in <a href="/facts/Henry_(unit)/uwVv9zue">henries</a>, and C is the <a href="/facts/Capacitance/0pH046TL">capacitance</a> in <a href="/facts/Farad/gPAYcVoE">farads</a>. The <a href="/facts/Angular_frequency/VqLm3L3R">angular frequency</a> 
 
 
 
 ω
 
 
 
 {\displaystyle \omega \,}
 
 has units of <a href="/facts/Radian/Srd38Sxa">radians</a> per second. A capacitor stores energy in the electric field between the plates, which can be written as follows:

U
          
            C
          
        
        =
        
          
            1
            2
          
        
        C
        
          V
          
            2
          
        
        =
        
          
            
              Q
              
                2
              
            
            
              2
              C
            
          
        
      
    
    {\displaystyle U_{C}={\frac {1}{2}}CV^{2}={\frac {Q^{2}}{2C}}}

Where Q is the net charge on the capacitor, calculated as

Q
        (
        t
        )
        =
        
          ∫
          
            −
            ∞
          
          
            t
          
        
        I
        (
        τ
        )
        d
        τ
      
    
    {\displaystyle Q(t)=\int _{-\infty }^{t}I(\tau )d\tau }

Likewise, an inductor stores energy in the magnetic field depending on the current, which can be written as follows:

U
          
            L
          
        
        =
        
          
            1
            2
          
        
        L
        
          I
          
            2
          
        
        =
        
          
            
              ϕ
              
                2
              
            
            
              2
              L
            
          
        
      
    
    {\displaystyle U_{L}={\frac {1}{2}}LI^{2}={\frac {\phi ^{2}}{2L}}}

Where 
 
 
 
 ϕ
 
 
 {\displaystyle \phi }
 
 is the branch flux, defined as

ϕ
        (
        t
        )
        ≡
        
          ∫
          
            −
            ∞
          
          
            t
          
        
        V
        (
        τ
        )
        d
        τ
      
    
    {\displaystyle \phi (t)\equiv \int _{-\infty }^{t}V(\tau )d\tau }

Since charge and flux are <a href="/facts/Canonical_coordinates/9g2SFFRz">canonically conjugate variables</a>, one can use <a href="/facts/Canonical_quantization/AKHX2JQn">canonical quantization</a> to rewrite the classical hamiltonian in the quantum formalism, by identifying

ϕ
        →
        
          
            
              ϕ
              ^
            
          
        
      
    
    {\displaystyle \phi \rightarrow {\hat {\phi }}}

q
        →
        
          
            
              q
              ^
            
          
        
      
    
    {\displaystyle q\rightarrow {\hat {q}}}

H
        →
        
          
            
              H
              ^
            
          
        
        =
        
          
            
              
                
                  
                    ϕ
                    ^
                  
                
              
              
                2
              
            
            
              2
              L
            
          
        
        +
        
          
            
              
                
                  
                    q
                    ^
                  
                
              
              
                2
              
            
            
              2
              C
            
          
        
      
    
    {\displaystyle H\rightarrow {\hat {H}}={\frac {{\hat {\phi }}^{2}}{2L}}+{\frac {{\hat {q}}^{2}}{2C}}}

and enforcing the canonical commutation relation

[
          
            
              
                
                  ϕ
                  ^
                
              
            
            ,
            
              
                
                  q
                  ^
                
              
            
          
          ]
        
        =
        i
        ℏ
      
    
    {\displaystyle \left[{\hat {\phi }},{\hat {q}}\right]=i\hbar }

Quantum LC circuit open-in-new

Quantum LC circuit