mecànica cuàntica

d_{x}[ [k+1]( f(x) [o( (1/(k+1)!)x^{k+1} )o] g(x) ) ] = ...

... [k]( d_{x}[f(x)] [o( (1/k!)x^{k} )o] d_{x}[g(x)] ) [o( (1/k!)x^{k} )o] ...

... d_{x}[f(x)] [o( (1/k!)x^{k} )o] d_{x}[g(x)]

int[ [k]( d_{x}[f(x)] [o( (1/k!)x^{k} )o] d_{x}[g(x)] ) ] d[x] = ...

... [k+1]( f(x) [o( (1/(k+1)!)x^{k+1} )o] g(x) ) [o( (1/(k+1)!)x^{k+1} )o] ...

... ( f(x) [o( (1/(k+1)!)x^{k+1} )o] g(x) )^{[o(x)o](-1)}

 

[k]( d_{x}[f(x)] [o( (1/k!)x^{k} )o] d_{x}[g(x)] ) = [k+1]( f(x) [o( (1/(k+1)!)x^{k+1} )o] g(x) )

Lagranià cuàntic de primera especie:

( h^{2}/(2ml) )·d_{x}[f(x)] = ( E_{n}+qA(x) )·f(x)

f(x) = [1]( ( (2ml)/h^{2} )·x [o(x)o] int[ ( E_{n}+qA(x) ) ] d[x] )

f(x) = [0]( ( (2ml)/h^{2} )·( E_{n}+qA(x) ) )

Lagranià cuàntic de segona especie:

( h^{2}/(2m) )·d_{xx}^{2}[f(x)] = ( E_{n}+qA(x) )·f(x)

f(x) = [2]( ( (2m)/h^{2} )^{(1/2)}·(1/2!)·x^{2} [o( (1/2!)·x^{2} )o] ...

... int[ int[ ( E_{n}+qA(x) )^{(1/2)} ] d[x] ] d[x] )

f(x) = [0]( ( (2m)/h^{2} )^{(1/2)}·( E_{n}+qA(x) )^{(1/2)} )

Lagranià cuàntic de hyper-espai:

( (l^{n}h^{2})/(2m) )·d_{x...x}^{n+2}[f(x)] = ( E_{n}+qA(x) )·f(x)

f(x) = [n+2]( ( (2m)/(l^{n}h^{2}) )^{(1/(n+2))}·(1/(n+2)!)·x^{n+2} [o( (1/(n+2)!)·x^{n+2} )o] ...

... int[ ...(n+2)... int[ ( E_{n}+qA(x) )^{(1/(n+2))} ] d[x] ...(n+2)... ] d[x] )

f(x) = [0]( ( (2m)/(l^{n}h^{2}) )^{(1/(n+2))}·( E_{n}+qA(x) )^{(1/(n+2))} )

Hamiltonià cuàntic:

ih·d_{t}[f(x,t)] = ( E_{n}+qA(x) )·f(x,t)

ih·d_{x}[f(x,t)]·d_{t}[x] = ( E_{n}+qA(x) )·f(x,t)

f(x,t) = [1]( ((-i)/h)·x [o(x)o] int[ (1/d_{t}[x]) ] d[x] [o(x)o] int[ ( E_{n}+qA(x) ) ] d[x] )

f(x,t) = [0]( ((-i)/h)·(1/d_{t}[x])·( E_{n}+qA(x) ) )

Hamiltonià cuàntic de camp constant:

ih·d_{t}[f(x,t)] = ( E_{n}+qgx )·f(x,t)

ih·d_{x}[f(x,t)]·d_{t}[x] = ( E_{n}+qgx )·f(x,t)

f(x,t) = [1]( ((-i)/h)·x [o(x)o] t [o(x)o] ( E_{n}x+qg·(1/2)·x^{2} ) )

f(x,t) = [0]( ((-i)/h)·( m/(qgt) )·( E_{n}+qgx ) )

(m/2)·d_{t}[x(t)]^{2} = E_{n}+qg·x(t)

x(t) = ( (qg)/m )·(1/2)·t^{2}+(-1)·( E_{n}/(qg) )

d_{t}[x(t)] = ( (qg)/m )·t

[[1]]( f(x) ) = f(x)

[[k]]( f(x) )^{(1/k)} = f(x)

[[k]]( s^{k}·f(x) ) = s·[[k]]( f(x) )

d_{x}[ [[k+1]]( f(x) ) ] = [[k]]( d_{x}[f(x)] )

int[ [[k]]( d_{x}[f(x)] ) ] d[x] = [[k+1]]( f(x) )

[[k+1]]( f(x) ) = [[k]]( d_{x}[f(x)] )

(m/2)·d_{t}[x(t)]^{2} = E_{n}+(a/2)·( x(t) )^{2}

x(t) = [[(1/2)]]( e^{(a/m)t}+(-1)·(2/a)·E_{n} )

d_{t}[x(t)] = [[(-1)(1/2)]]( (a/m)·e^{(a/m)t} )

[[(-1)]]( (a/m)^{2}·e^{(a/m)t} ) = e^{(a/m)t}

Hamiltonià cuàntic de oscilador harmónic:

ih·d_{t}[f(x,t)] = ( E_{n}+(a/2)·x^{2} )·f(x,t)

ih·d_{x}[f(x,t)]·d_{t}[x] = ( E_{n}+(a/2)·x^{2} )·f(x,t)

f(x,t) = [1]( ((-i)/h)·x [o(x)o] t [o(x)o] ( E_{n}x+(a/6)·x^{3} ) )

f(x,t) = [0]( ((-i)/h)·( [[(1/2)]]( (a/m)·e^{(a/m)t} ) )·( E_{n}+(a/2)·x^{2} ) )