3D harmonski oscilator in vrtilna količina

Gregor Tavčar

Hamiltonova funkcija za sferno simetričen 3-D harmonski oscilator izražena v kartezičnih koordinatah je preproste oblike:

\(H = \(p\_x\^2 + p\_y\^2 + p\_y\^2\)\/\(2  m\) + \(\(m\ ω\^2\)\/2\) \((x\^2 + y\^2 + z\^2)\)\)
\(H\&^ = \(\(-ℏ\)\/\(2  
                m\)\) \((∂\^2\/∂x\^2 + ∂\^2\/∂y\^2 + ∂\^2\/∂\ z\^2)\) + \(\(m\ ω\^2\)\/2\) \((x\^2 + y\^2 + z\^2)\)\)

S kreacijskim in anihilacijskim operatorjem

\(a\_w\^† = \(\(1\/\@2\) \((w\/w\_0 - \(w\_0\) ∂\/∂w)\)\(\ \)\)\)
(Subscript[a, w]) = (1/(Sqrt[2])) ((w/(Subscript[w, 0])) + (Subscript[w, 0]) (∂/∂)w)
\(w = x, y, 
    z\ \ \ \ \ \ \ w\_0 = \(x\_0 = \(y\_0 = \(z\_0 = \@\(ℏ\/\(ω\ m\)\)\)\)\)\)

zapišemo operator hamiltonove funkcije kot:

\(H\&^ = \(\(-ℏ\)\/\(2  
              m\)\) \((\((\(a\_x\^†\) a\_x + 
                1\/2)\) + \((\(a\_y\^†\) a\_y + 
                1\/2)\) + \((\(a\_z\^†\) a\_z + 
                1\/2)\))\)\)

Vidimo, da je zgornja hamiltonka sestavljena iz členov, ki vsak zase predstavljajo hamiltonko enodimenzionalnega harmonskega oscilatorja, zato lahko lastne funkcije  hamiltonke 3-D harmonskega oscilatorja ψ(x,y,z) zapišemo kot produkt lastnih funkcij hamiltonke 1-D harmonskega oscilatorja

ψ (x, y, 
          z) = ξ (x) η (y) ζ (z)

Preverimo, da je zgornji ψ res lastna funkcija za našo hamiltonko in poiščimo njene lastne vrednosti:

\(\(H\&^\) ψ\  = \(H\&^\ ξηζ\  = \(\(\(-ℏ\)\/\(2  
                    m\)\) \((\(∂\^2 ξ\/∂x\^2\) ηζ + \(∂\^2 η\/∂y\^2\) ξζ + \(∂\^2 ζ\/∂\ 
                          z\^2\) ξη)\) + \(\(m\ ω\^2\)\/2\) \((x\^2 + y\^2 + 
                  z\^2)\) ξηζ = 
\(\(=\)\(\(\(-ℏ\)\/\(2  
                      m\)\) \((\((\(a\_x\^†\) \(a\_x\) ξ + \(1\/2\) ξ)\) ηζ + \((\(a\_y\^†\) \(a\_y\) η + \(1\/2\) η)\) ξζ + \((\(a\_z\^†\) \(a\_z\) ζ + \(1\/2\) ζ)\) ξη)\) = 
\(\(=\)\(ηζ\ \(E\_x\) ξ\  + \ ξζ\ \(E\_y\) η\  + \ ξη\ \(E\_z\) ζ\  = \(\((E\_x + E\_y + 
                          E\_z)\) ξηζ = 
\(\(=\)\(\ \ \)\(ℏ\ ω\ \((n\_x + n\_y + 
                            n\_z + 
                            3\/2)\) ψ\)\)\)\)\)\)\)\)\)\)

V vektorskem zapisu bomo lastno funkcijo pisali kot:

\(ψ = \(\(\(|\)\(\(n\_x\)\(>\)\)\(|\)\(\(n\_y\)\(>\)\)\(\ \)\(|\)\(\(n\_z\)\(>\)\)\)\  = \ \(\(|\)\(n\_x\)\)\), 
    n\_y, \(\(n\_z\)\(>\)\)\)

Delujmo na zgornjo 3-D lastno funkcijo z 1-D operatorji:

(Subscript[a, x]) | (Subscript[n, x]), (Subscript[n, y]), (Subscript[n, z])>  =  (Sqrt[(Subscript[n, x])]) | (Subscript[n, x]) - 1, 
    (Subscript[n, y]), (Subscript[n, z])>;
((Subscript[a, x])^†) | (Subscript[n, x]), 
    (Subscript[n, y]), (Subscript[n, z])>  =  (Sqrt[(Subscript[n, x]) + 1]) | (Subscript[n, x]) + 1, 
    (Subscript[n, y]), (Subscript[n, z])>;
\(\(a\_x\^†\) a\_x | n\_x, 
    n\_y, \(\(n\_z\)\(>\)\)\  = \(n\&^\)\_x | n\_x, 
    n\_y, \(\(n\_z\)\(>\)\)\  = \ n\_x | n\_x, 
    n\_y, \(\(n\_z\)\(>\)\)\)

(\(\(n\&^\)\_x\) je operator štetja, (Subscript[n, x]) pa njegova lastna vrednost) Podobno velja za 1-D operatorje drugih dveh koordinatnih osi.

Zapišimo operatorje vrtilne količine našega 3-D harmonskega oscilatorja s kreacijskim in anihilacijskim operatorjem (izpeljava ostrega pogleda, ki se lepše bere v nasprotni smeri)

\(\(L\&^\)\_z = \(\(-i\)\ ℏ\ \((x ∂\/∂y - 
              y ∂\/∂x)\)\  = 
\(\(=\)\(\(-\(\(i\ ℏ\)\/2\)\) \((\(x\ y\)\/\(\(x\_0\) y\_0\) + \(y\_0\/x\_0\) 
                    x ∂\/∂y - \(x\_0\/y\_0\) 
                    y ∂\/∂x - \(y\_0\) \(x\_0\) ∂\^2\/\(∂x ∂y\) - \((\(x\ y\)\/\(\(x\_0\) y\_0\) + \(x\_0\/y\_0\) y ∂\/∂x - \(y\_0\/x\_0\) 
                        x ∂\/∂y - \(y\_0\) \(x\_0\) ∂\^2\/\(∂x ∂y\))\))\) = 
\(\(=\)\(\(-i\)\ ℏ \((\(1\/2\) \((x\/x\_0 - \(x\_0\) ∂\/∂x)\) \((y\/y\_0 + \(y\_0\) ∂\/∂y)\) - \(1\/2\) \((y\/y\_0 - \(y\_0\) ∂\/∂y)\) \((x\/x\_0 + \(x\_0\) ∂\/∂x)\))\) = 
\(\(=\)\(\ \)\(\(-i\)\ ℏ \((\(a\_x\^†\) a\_y - \(a\_y\^†\) 
                          a\_x)\)\)\)\)\)\)\)\)\)

\(\(L\&^\)\_y = \(-i\)\ ℏ \((\(a\_z\^†\) 
              a\_x - \(a\_x\^†\) a\_z)\)\)
\(\(L\&^\)\_x = \(-i\)\ ℏ \((\(a\_y\^†\) 
              a\_z - \(a\_z\^†\) a\_y)\)\)

Zapišimo še operator kvadrata vrtilne količine z anihilacijskim in kreacijskim operatorjem (upoštevamo, da enodimenzionalna operatorja različnih koordinatnih osi med seboj komutirata):

[(Subscript[a, w]), ((Subscript[a, w])^†)] =  1
\(\(L\&^\)\^2 = \(\(L\&^\)\_x\%2 + \(L\&^\)\_y\%2 + \(L\&^\)\_z\%2\  = \ \(-\ ℏ\^2\) \((\((\(a\_y\^†\) a\_z - \(a\_z\^†\) a\_y)\)\^2 + \((\(a\_z\^†\) a\_x - \(a\_x\^†\) a\_z)\)\^2 + \((\(a\_x\^†\) a\_y - \(a\_y\^†\) a\_x)\)\^2)\)\)\)
\(\(L\&^\)\_z\%2 = \(\(-ℏ\^2\) \((\(a\_x\^†\) \(a\_x\^†\) \(a\_y\) a\_y - \(a\_y\^†\) \(a\_y\) \(a\_x\) 
                a\_x\^† - \(a\_x\^†\) \(a\_x\) \(a\_y\) 
                a\_y\^† + \(a\_y\^†\) \(a\_y\^†\) \(a\_x\) a\_x)\) = 
\(\(=\)\(\(-ℏ\^2\) \((\(a\_x\^†\) \(a\_x\^†\) \(a\_y\) 
                    a\_y + \(a\_y\^†\) \(a\_y\^†\) \(a\_x\) 
                    a\_x - \(a\_y\^†\) \(a\_y\) \((\(a\_x\^†\) a\_x + 
                        1)\) - \(a\_x\^†\) \(a\_x\) \((\(a\_y\^†\) a\_y + 
                        1)\))\) = 
\(\(=\)\(ℏ\^2\ \((\(a\_x\^†\) a\_x + \(a\_y\^†\) a\_y + 
                    2 \( a\_y\^†\) \(a\_y\) \(a\_x\^†\) 
                      a\_x - \(a\_x\^†\) \(a\_x\^†\) \(a\_y\) 
                      a\_y - \(a\_y\^†\) \(a\_y\^†\) \(a\_x\) 
                      a\_x)\)\)\)\)\)\)\)

\(\(L\&^\)\_y\%2 = \ \(ℏ\^2\) \((\ \(a\_z\^†\) 
              a\_z + \(a\_x\^†\) a\_x + 
            2 \( a\_x\^†\) \(a\_x\) \(a\_z\^†\) 
              a\_z - \(a\_z\^†\) \(a\_z\^†\) \(a\_x\) 
              a\_x - \(a\_x\^†\) \(a\_x\^†\) \(a\_z\) 
              a\_z)\)\)

\(\(L\&^\)\_x\%2 = \ \(ℏ\^2\) \((\(a\_y\^†\) 
              a\_y + \(a\_z\^†\) a\_z + 
            2 \( a\_z\^†\) a\_z\ \(a\_y\^†\) 
              a\_y - \(a\_y\^†\) \(a\_y\^†\) \(a\_z\) 
              a\_z - \(a\_z\^†\) \(a\_z\^†\) \(a\_y\) 
              a\_y)\)\)

\(\(L\&^\)\^2 = \ \(ℏ\^2\) \((\((\(a\_y\^†\) 
                  a\_y + \(a\_z\^†\) a\_z + 
                2 \( a\_z\^†\) a\_z\ \(a\_y\^†\) 
                  a\_y - \(a\_y\^†\) \(a\_y\^†\) \(a\_z\) 
                  a\_z - \(a\_z\^†\) \(a\_z\^†\) \(a\_y\) 
                  a\_y)\) + 
\((\ \(a\_z\^†\) 
                  a\_z + \(a\_x\^†\) a\_x + 
                2 \( a\_x\^†\) \(a\_x\) \(a\_z\^†\) 
                  a\_z - \(a\_z\^†\) \(a\_z\^†\) \(a\_x\) 
                  a\_x - \(a\_x\^†\) \(a\_x\^†\) \(a\_z\) 
                  a\_z)\) + 
\((\(a\_x\^†\) 
                  a\_x + \(a\_y\^†\) a\_y + 
                2 \( a\_y\^†\) \(a\_y\) \(a\_x\^†\) 
                  a\_x - \(a\_x\^†\) \(a\_x\^†\) \(a\_y\) 
                  a\_y - \(a\_y\^†\) \(a\_y\^†\) \(a\_x\) 
                  a\_x)\))\)\)

Delujmo z operatorjem kvadrata vrtilne količine na nekaj lastnih funkcij hamiltonke z najnižjimi energijami:

\(\(L\&^\)\^2 | 1, 0, \(\(0\)\(>\)\)\  = \(L\&^\)\_x\%2 | 1, 0, 
    0 > \(+\(L\&^\)\_y\%2\) | 1, 0, 0 > \(+\(L\&^\)\_z\%2\) | 1, 
    0, \(\(0\)\(>\)\)\  = 
\(\(=\)\(\(ℏ\^2\) \((\((\(a\_y\^†\) a\_y | 1, 0, 0 > \(+a\_z\^†\) a\_z | 1, 0, 
                  0 > \(+2\) \(a\_z\^†\) a\_z\ \(a\_y\^†\) 
                        a\_y | 1, 0, 
                  0 > \(-a\_y\^†\) \(a\_y\^†\) \(a\_z\) 
                        a\_z | 1, 0, 
                  0 > \(-a\_z\^†\) \(a\_z\^†\) \(a\_y\) 
                        a\_y | 1, 
                  0, \(\(0\)\(>\)\))\) + 
\((\ \(a\_z\^†\) a\_z | 1, 0, 0 > \(+a\_x\^†\) a\_x | 1, 0, 
                  0 > \(+2\) \(a\_x\^†\) \(a\_x\) \(a\_z\^†\) 
                        a\_z | 1, 0, 
                  0 > \(-a\_z\^†\) \(a\_z\^†\) \(a\_x\) 
                        a\_x | 1, 0, 
                  0 > \(-a\_x\^†\) \(a\_x\^†\) \(a\_z\) 
                        a\_z | 1, 
                  0, \(\(0\)\(>\)\))\) + 
\((\(a\_x\^†\) a\_x | 1, 0, 0 > \(+a\_y\^†\) a\_y | 1, 0, 
                  0 > \(+2\) \(a\_y\^†\) \(a\_y\) \(a\_x\^†\) 
                        a\_x | 1, 0, 
                  0 > \(-a\_x\^†\) \(a\_x\^†\) \(a\_y\) 
                        a\_y | 1, 0, 
                  0 > \(-a\_y\^†\) \(a\_y\^†\) \(a\_x\) 
                        a\_x | 1, 
                  0, \(\(0\)\(>\)\))\))\) = 
\(\(=\)\(\(ℏ\^2\) \((\((0 + 0 + 0 + 0 + 0)\) + \((\(\(0\)\(+\)\) | 1, 0, 
                      0 > \(+0\) + 0 + 0)\) + \((\(\(|\)\(1\)\), 0, 
                      0 > \(+0\) + 0 + 0 + 
                          0)\))\) = 
\(\(=\)\(2  ℏ\^2 | 1\)\)\)\)\)\), 
    0, \(\(0\)\(>\)\)\  = 
		\(\(=\)\(l \((l + 
                1)\) ℏ\^2 | 1\)\), 0, \(\(0\)\(>\)\), 
    l = 1
\)
\(\(L\&^\)\^2 | 0, 1, \(\(0\)\(>\)\)\  = 2  ℏ\^2 | 0, 
    1, \(\(\(0\)\(>\)\);\)\)
\(\(L\&^\)\^2 | 0, 0, \(\(1\)\(>\)\)\  = 2  ℏ\^2 | 0, 
    0, \(\(\(1\)\(>\)\);\)\)
\(\(L\&^\)\^2 | 0, 1, \(\(1\)\(>\)\)\  = 6  ℏ\^2 | 0, 
    1, \(\(\(1\)\(>\)\);\)\)
\(\(L\&^\)\^2 | 1, 1, \(\(0\)\(>\)\)\  = 6  ℏ\^2 | 1, 
    1, \(\(\(0\)\(>\)\);\)\)
\(\(L\&^\)\^2 | 1, 0, \(\(1\)\(>\)\)\  = 6  ℏ\^2 | 1, 
    0, \(\(\(1\)\(>\)\);\)\)
\(\(L\&^\)\^2 | 1, 1, \(\(1\)\(>\)\)\  = 12  ℏ\^2 | 1, 
    1, \(\(\(1\)\(>\)\);\)\)
\(\(L\&^\)\^2 | 0, 0, \(\(0\)\(>\)\)\  = \(0 = 0*ℏ\^2 | 0\), 
    0, \(\(0\)\(>\)\)\)

Vidimo torej, da so zgornje lastne funkcije hamiltonke tudi lastne funkcije operatorja kvadrata vrtilne količine, sepravi imajo doro določen kvadrat vrtilne količine- prve tri imajo l=1, druge tri l=2, predzadnja l=3 in zadnja l=0 (|2,0,0> pa že nima več ostro določenega kvadrata vrtilne količine - preveri!).

Poglejmo še kako zapišemo stanje z dobro določeno vrtilno količino v z smeri (označimo jih z γ)  z lastnimi funkcijami hamiltonke  (zanimajo nas samo stanja z l=1):

\(\(\(L\&^\)\_z\) γ = ℏ\ m\ γ, \ m = \(-1\), 0, 
    1\)
\(\(\(L\&^\)\_z\) γ\_\(m = 1\) = \(ℏ\ *1*\((A | 1, 0, 
            0 > \(+B\) | 0, 1, 0 > \(+C\) | 0, 
            0, \(\(1\)\(>\)\))\)\  = 
\(\(=\)\(\(-i\)\ ℏ \((\(a\_x\^†\) a\_y - \(a\_y\^†\) a\_x)\) \((A | 1, 
                0, 0 > \(+B\) | 0, 1, 0 > \(+C\) | 0, 
                0, \(\(1\)\(>\)\))\) = 
\(\(=\)\(\(-i\)\ ℏ \((\(-A\) | 0, 1, 0 > \(+B\) | 1, 0, 
                      0 > \(+C\)*0)\)⟹
⟹\ i\ A = B\)\)\)\)\), \(-i\)\ B = 
      A, \ C = \(0\ ⟹A = i\/\@2\), B = \(-1\)\/\@2, 
    C = 0\)
\(γ\_\(m = 1\) = \((i\/\@2 | 1, 0, 
        0 > \(-\(1\/\@2\)\) | 0, 
        1, \(\(0\)\(>\)\))\), 
γ\_\(m = 0\) = \(\(|\)\(0\)\), 
    0, \(\(1\)\(>\)\), 

γ\_\(m = \(-1\)\) = \((1\/\@2 | 1, 0, 0 > \(-\(i\/\@2\)\) | 0, 
        1, \(\(0\)\(>\)\))\)\)