Lectures on Quantum Mechanics

1 day ago by reibaretti

#Mathematical Methods/Methods of Theoretical Physics #http://www1.uprh.edu/rbaretti #http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysics.htm # Lectures on Quantum Mechanics http://www1.uprh.edu/rbaretti/LQMIntro.htm # http://www1.uprh.edu/rbaretti/LQMch1.htm #http://www1.uprh.edu/rbaretti/LQMch2.htm # http://www1.uprh.edu/rbaretti/LQMch3.htm # http://www1.uprh.edu/rbaretti/LQMch4.htm # http://www1.uprh.edu/rbaretti/LQMch5.htm # http://www1.uprh.edu/rbaretti/LQMch6.htm #.. # http://www1.uprh.edu/rbaretti/LQMintro.htm diff(x^3,x,2) 
        
6*x
f=x^2 diff(f,x) 
        
2*x
integral(f,x) 
        
x^3/3
integral(f,x,0,3) 
        
9
#x,y=var('x,y') # plot3d(16-x^2-y^2,(x,-2,2),(y,-2,2)) 
       
#y=plot(sin(x),0,pi) #show(y) 
       
y=plot(exp(-x),0,4) show(y) 
       
plot ((x^2+1),x,-2,2).show(xmin=-3,xmax=3,ymin=0.,ymax=6) 
       
integral(exp(-2*x),x,0,20) 
        
1/2 - e^-40/2
# two linear equations x, y = var('x, y') solve([8*x + y == 3, -x +7*y == 0],x,y) 
        
[[x == 7/19, y == 1/19]]
# example of Kirchoff eqs fro three currents i1,i2,i3=var('i1,i2,i3') solve([8*i1 + i2 +i3== 3, -i1 +7*i2 +i3== 0,i1+i2+i3==1],i1,i2,i3) 
        
[[i1 == 2/7, i2 == -1/14, i3 == 11/14]]
# To solve the equation x''+9*x=0: t = var('t') # define a variable t x = function('x',t) # define x to be a function of that variable DE = lambda y: diff(y,t,2) +9*y desolve(DE(x(t)), [x,t]) 
        
k1*sin(3*t) + k2*cos(3*t)
# To solve the equation x''+ 9*x=0: t = var('t') # define the variable t x=function('x',t) DE=lambda x:diff(x,t,2)+diff(x,t)+ 9*x desolve (DE(x(t)),[x,t]) 
        
e^(-(t/2))*(k1*sin(sqrt(35)*t/2) + k2*cos(sqrt(35)*t/2))
var('x,y,k ,L') x=10 f=2*k*L/(x^2+y^2)^(1/2) integral(f,y,0,100*x) 
        
2*arcsinh(100)*k*L
# the R declaration defines the decimal result in terms of bits R=RealField(30) f=1/(5+x^2)^(1/2) g=integral(f,x,0,5) R(integral(f,x,0,5)) 
        
1.5444850
var('x, y') y = integral(sqrt(1 + x^2), x, 0, 2) print y # n(y,20) 
        
arcsinh(2) + 2 sqrt(5)
                             --------------------
                                      2
n((arcsinh(2) + 2*sqrt(5))/2,20) 
        
2.9579
# 2x2 matrix A = matrix(2, 2,[1/(3), -1/(3), 1/(3), 1/(3)]); print(A) 
        
Syntax Error:
     A = matrix(2, 2,[1/(3), -1/(3), 1/(3), 1/(3)]);
plot (.434*log(x),x,1,10).show(xmin=0,xmax=10,ymin=0.,ymax=1) 
       
a=pi;p=3; y=plot((a/pi)^(1/4)*cos(p*x)*exp(-a*x^2/2),-3,3) show(y) #cos(px/h') (α/π )1/4 exp ( - αx2 /2) 
       
# Ψ(x) = (α/π)1/4 exp( i k0 x) exp[-(α/2) (x-x0)2 alfa=1 ; k0=1; x0=1; repsi=(alfa/pi)^(1/4)*cos(k0*x)*exp(-(alfa/2)*(x-x0)^2 ); impsi=(alfa/pi)^(1/4)*sin(k0*x)*exp(-(alfa/2)*(x-x0)^2 ); #y= plot(repsi,x,-4*x0,5*x0) #y= plot(impsi,x,-4*x0,5*x0) y= plot(repsi^2+impsi^2,x,-4*x0,5*x0) show(y) 
       
var('x,p,a');a=2;p=1; f= cos(p*x)*exp(-a*x^2); integral(f,x,0,oo) 
        
1/4*sqrt(pi)*sqrt(2)*e^(-1/8)
R=RealField(20) #R(2/75) #R( sqrt(pi)*e^(-1/4) ) R( 1/2*sqrt(pi)*erf(50) ) 
        
0.88623
integral(exp(-x),x,0,oo) 
        
1
f=x*( sin(x)^2+(1/4)*sin(3*x)^2 ); integral(f,x,0,2*pi) 
        
5/4*pi^2
f=sin(x)^2+(1/4)*sin(3*x)^2 ; integral(f,x,0,2*pi) 
        
5/4*pi
f=x^2*( sin(x)^2+(1/4)*sin(3*x)^2 ); integral(f,x,0,2*pi) 
        
-37/72*pi + 5/3*pi^3
R= RealField(20) R ( (-37*pi)/72 + (5/3)*pi^3 )*(4/(5*pi) ) 
        
40.050/pi
 
       
f=sin(x)*cos(3*x) ; integral(f,x,0,2*pi) 
        
0
# { sin(x)2 +(9/4)sin(3x)2 } dx /(5π/4) f=(4/(5*pi))*(sin(x)^2 +(9/4)*sin(3*x)^2 ) ; integral(f,x,0,2*pi) 
        
13/5
psi=exp(-x^2/2) ; v=(1/2)*x^2; -(1/2)*diff(psi,x,2)+v*psi 
        
1/2*e^(-1/2*x^2)
f=x*(1-x); integral(f,x,0,1) 
        
1/6
psi=exp(-x^2) ; v=(1/2)*x^2; -(1/2)*diff(psi,x,2)+v*psi 
        
-3/2*x^2*e^(-x^2) + e^(-x^2)
f=(30)^.5*x*(1-x); integral(f^2,x,0,1) 
        
1.0
var('u');f=1/(exp(u)-1); y=plot(f,u,.1,2) show(y) 
       
var('x');k1=1; k2=1 ; f=cos(k1*x) - (k2/k1)*sin(k1*x); # f=exp(-k2*x); y=plot(f,x,-7,0) show(y) 
       
var('x');k1=1; k2=1 ; f=exp(-k2*x); y=plot(f,x,0,2) show(y) 
       
e^(I)-e^(-I) 
        
-e^(-I) + e^I
var('x') mu=0 ; sigma=1.e-3 ;eta=10; gauss= (1/sigma)*(1/(2*pi)^(1/2))*exp(-(x-mu)^2/(2*sigma^2)); v=-eta*gauss; #v =x; y=plot(v,x,-3*sigma,3*sigma) show(y) 
       
v=x y=plot(v,x,-3,3) show(y) 
       
# creation and annihilation operators for the harmonic oscillator f0=exp(-x^2/2); -diff(f0,x)+x*f0 
        
2*x*e^(-1/2*x^2)
f1=2*x*e^(-1/2*x^2);-diff(f1,x)+x*f1 
        
4*x^2*e^(-1/2*x^2) - 2*e^(-1/2*x^2)
f0=exp(-x^2/2); diff(f0,x)+x*f0 
        
0
 
       
f1=2*x*e^(-1/2*x^2); diff(f1,x)+x*f1 
        
2*e^(-1/2*x^2)
f=2*x*exp(-x^2/2); d2= -(1/2)*diff(f,x,2); v=(1/2)*x^2*f; d2+v 
        
3*x*e^(-1/2*x^2)
f=(2/pi^(1/2))*x^2*exp(-x^2/2); integral(f,x,-oo,oo) 
        
2*sqrt(2)
f=exp(-x^2/2); -(1/2)*diff(f,x,2) +(1/2)*x^2*f 
        
1/2*e^(-1/2*x^2)
f=(1/pi)^(1/4)*exp(-x^2/2); #f=exp(-x^2/2); integral(f^2,x,-oo,+oo) 
        
1
f=(1/pi)^(1/4)*exp(-x^2/2); -(1/2)*diff(f,x,2) +(1/2)*x^2*f 
        
1/2*e^(-1/2*x^2)/pi^(1/4)
f=(1/pi)^(1/4)*exp(-x^2/2); y=plot(f,x,-3,3) show(y) 
       
f0=(1/pi)^(1/4)*exp(-x^2/2); (1/2^(1/2))*(-diff(f0,x)+x*f0) 
        
sqrt(2)*x*e^(-1/2*x^2)/pi^(1/4)
f1=sqrt(2)*x*e^(-1/2*x^2)/pi^(1/4); #integral(f1^2,x,-oo,+oo) (1/2^(1/2))*(-diff(f1,x)+x*f1) 
        
1/2*(2*sqrt(2)*x^2*e^(-1/2*x^2)/pi^(1/4) -
sqrt(2)*e^(-1/2*x^2)/pi^(1/4))*sqrt(2)
f2=1/2*(2*sqrt(2)*x^2*e^(-1/2*x^2)/pi^(1/4) - sqrt(2)*e^(-1/2*x^2)/pi^(1/4))*sqrt(2) integral(f2^2,x,-oo,+oo) 
        
2
f2=1/2*(2*sqrt(2)*x^2*e^(-1/2*x^2)/pi^(1/4) - sqrt(2)*e^(-1/2*x^2)/pi^(1/4))*sqrt(2) (1/2^(1/2))*(-diff(f2,x)+x*f2) 
        
1/4*((2*sqrt(2)*x^2*e^(-1/2*x^2)/pi^(1/4) -
sqrt(2)*e^(-1/2*x^2)/pi^(1/4))*sqrt(2)*x +
(2*sqrt(2)*x^3*e^(-1/2*x^2)/pi^(1/4) -
5*sqrt(2)*x*e^(-1/2*x^2)/pi^(1/4))*sqrt(2))*sqrt(2)
f3=1/4*((2*sqrt(2)*x^2*e^(-1/2*x^2)/pi^(1/4) - sqrt(2)*e^(-1/2*x^2)/pi^(1/4))*sqrt(2)*x + (2*sqrt(2)*x^3*e^(-1/2*x^2)/pi^(1/4) - 5*sqrt(2)*x*e^(-1/2*x^2)/pi^(1/4))*sqrt(2))*sqrt(2) integral(f3^2,x,-oo,+oo) 
        
6
f1=sqrt(2)*x*e^(-1/2*x^2)/pi^(1/4); integral(f1*(-(1/2)*diff(f1,x,2)+(1/2)*x^2*f1),x,-oo,+oo) 
        
3/2
f2=1/2*(2*sqrt(2)*x^2*e^(-1/2*x^2)/pi^(1/4) - sqrt(2)*e^(-1/2*x^2)/pi^(1/4))*sqrt(2); #integral(f2*(-(1/2)*diff(f2,x,2)+(1/2)*x^2*f2),x,-oo,+oo) integral(f2^2,x,-oo,+oo) 
        
2
# (1/2^(1/2) ) ( - d/dx + x ) (1/π)1/4 exp(-x2 /2) psi0= (1/pi^(1/4))*exp(-x^2/2); psiplus=(1/2^(1/2) )*(-diff(psi0,x)+x*psi0 ); integral(psiplus^2,x,-oo,+oo) 
        
1
psiplus=sqrt(2)*x*e^(-1/2*x^2)/pi^(1/4); hpsiplus= (-1/2)*diff(psiplus,x,2) +(1/2)*x^2*psiplus; integral(psiplus*hpsiplus,x,-oo,+oo) 
        
3/2
f=sqrt(2)*x*e^(-1/2*x^2)/pi^(1/4); y=plot(f,x,-4,4) show(y) 
       
# creation operator a+ psi1 = f=sqrt(2)*x*e^(-1/2*x^2)/pi^(1/4); (1/2^(1/2))*(-diff(f,x) +x*f) 
        
1/2*(2*sqrt(2)*x^2*e^(-1/2*x^2)/pi^(1/4) -
sqrt(2)*e^(-1/2*x^2)/pi^(1/4))*sqrt(2)
#1/2*( 2*sqrt(2)*x^2*e^(-1/2*x^2)/pi^(1/4) - #sqrt(2)*e^(-1/2*x^2)/pi^(1/4) )*sqrt(2) psiplus= (1/pi^(1/4))*(2*x^2 -1)*exp(-x^2/2); integral(psiplus^2,x,-oo,+oo) 
        
2
psi2=(1/2^(1/2))*(1/pi^(1/4))*(2*x^2 -1)*exp(-x^2/2); integral(psi2^2,x,-oo,+oo) 
        
1
psi2=(1/2^(1/2))*(1/pi^(1/4))*(2*x^2 -1)*exp(-x^2/2); integral(psi2*( (-1/2)*diff(psi2,x,2) +(1/2)*x^2*psi2),x,-oo,+oo) 
        
5/2
psi2=(1/2^(1/2))*(1/pi^(1/4))*(2*x^2 -1)*exp(-x^2/2); psi3=(1/3^(1/2))*(1/2^(1/2))*(-diff(psi2,x) +x*psi2); #integral(psi3^2,x,-oo,+oo) # 1 integral(psi3*( (-1/2)*diff(psi3,x,2) +(1/2)*x^2*psi3),x,-oo,+oo) # 7/2 
        
7/2
psi2=(1/2^(1/2))*(1/pi^(1/4))*(2*x^2 -1)*exp(-x^2/2); psi3=(1/3^(1/2))*(1/2^(1/2))*(-diff(psi2,x) +x*psi2); integral(psi3^2,x,-oo,oo) 
        
1
psi3=(1/3^(1/2))*(1/2^(1/2))*(-diff(psi2,x) +x*psi2); psi3 
        
1/6*((2*x^2 - 1)*sqrt(2)*x*e^(-1/2*x^2)/pi^(1/4) -
2*sqrt(2)*x*e^(-1/2*x^2)/pi^(1/4))*sqrt(2)*sqrt(3)
psi3=(1/6)*2*sqrt(3)*(exp(-1/2*x^2)/pi^(1/4))*(2*x^3-3*x); integral(psi3^2,x,-oo,oo) 
        
1
psi3= (1/3^(1/2) )*(1/pi^(1/4))*exp( -x^2/2)*( 2*x^3 - 3*x ); #integral(psi3^2,x,-oo,oo) d2=-(1/2)*diff(psi3,x,2) integral(psi3*(d2+(1/2)*x^2*psi3),x,-oo,oo) 
        
7/2
psi0=(1/pi^(1/4))*exp(-x^2/2) ; (1/2^(1/2))*(diff(psi0,x)+x*psi0) 
        
0
psi1=(2^(1/2)/pi^(1/4))*x*exp(-x^2/2); (1/2^(1/2))*(diff(psi1,x)+x*psi1) 
        
e^(-1/2*x^2)/pi^(1/4)
psi2 =(1/2^(1/2))* (1/pi^(1/4) )*(2*x^2 -1)*exp(-x^2/2); (1/2^(1/2))*(diff(psi2,x)+x*psi2) 
        
2*x*e^(-1/2*x^2)/pi^(1/4)
# normalization of harmonic oscillator eigenfunctions h0=1; psi0=h0*exp(-x^2/2); s0=integral(psi0^2,x,-oo,oo); n0=1/s0^(1/2); s0 , n0 
        
(sqrt(pi), pi^(-1/4))
h1=2*x; psi1=h1*exp(-x^2/2); s1=integral(psi1^2,x,-oo,oo); n1=1/s1^(1/2); s1 , n1 
        
(2*sqrt(pi), 1/2*sqrt(2)/pi^(1/4))
h2=4*x^2-2; psi2=h2*exp(-x^2/2); s2=integral(psi2^2,x,-oo,oo); n2=1/s2^(1/2); s2 , n2 
        
(8*sqrt(pi), 1/4*sqrt(2)/pi^(1/4))
h3= 8*x^3-12*x ; psi3=h3*exp(-x^2/2); s3=integral(psi3^2,x,-oo,oo); n3=1/s3^(1/2); s3 , n3 
        
(48*sqrt(pi), 1/12*sqrt(3)/pi^(1/4))
# plot periodic potential A=-10 ; L=1 ; v=A*sin(2*pi*x/L)^2; y=plot(v,x,0,5*L) show(y) 
       
# plot periodic potential v0=10 ; d=1 ; v=v0*(1-sin(pi*x/d)^2) ; y=plot(v,x,0,6*d) show(y) 
       
var('t,x') A=2; w=2*pi ; k=2*pi;c=w/k; x=1;tau=2*pi/w; psi=A*sin(w*t)*cos(k*(x+c*t)); y=plot(psi,t,0,3*tau); show(y) 
       
var('r'); L=1 ; f= (L^2/(2*r^2) -1/r) ; y=plot(f,r,0.4,4); show(y) 
       
sage: x = PolynomialRing(QQ, 'x').gen() sage: y = PolynomialRing(QQ, 'y').gen() sage: spherical_harmonic(3,2,x,y) '15*sqrt(7)*cos(x)*sin(x)^2*e^(2*I*y)/(4*sqrt(30)*sqrt(pi))' sage: spherical_harmonic(3,2,1,2) 
        
1/8*sqrt(7)*sqrt(30)*e^(4*I)*sin(1)^2*cos(1)/sqrt(pi)
var('x') psi1= (30)^(1/2)*x*(1-x); psi2= 2*(210)^(1/2)*( x^2*(1-x) -(1-x)*x/2); psi3=(17640)^(1/2)*( x^3*(1-x)-((30)^(1/2)/105 )*psi1  -(1/( 2(210)^(1/2) ))*psi2 ) ; c1=1 ; c3=-0.0380019285; phi1=c1*x^2; integral(phi^2,x,0,1) 
        
Syntax Error:
    c1=1 ;
var('r'); f=( 2-r)*exp(-r/2); integral(4*pi*r^2*f^2,r,0,oo) 
        
32*pi
var('r');f1=exp(-r); f2=( 2-r)*exp(-r/2);f3=(a+b*r+r^2)*exp(-r/3); integral(r^2*f1*f3,r) 
        
-3/32*(8*r^2 + 12*r + 9)*a*e^(-4/3*r) - 3/128*(32*r^3 + 72*r^2 +
108*r + 81)*b*e^(-4/3*r) - 3/128*(32*r^4 + 96*r^3 + 216*r^2 + 324*r
+ 243)*e^(-4/3*r)
f=a*x; g=integral(f,x); x=4;g 
        
1/2*a*x^2
x=4 (1/2)*a*x^2 
        
8*a
var('r,a,b');f1=exp(-r); f2=( 2-r)*exp(-r/2);f3=(a+b*r+r^2)*exp(-r/3); integral(r^2*f2*f3,r) 
        
-12/125*(25*r^2 + 60*r + 72)*a*e^(-5/6*r) + 6/625*(125*r^3 + 450*r^2
+ 1080*r + 1296)*a*e^(-5/6*r) - 12/625*(125*r^3 + 450*r^2 + 1080*r +
1296)*b*e^(-5/6*r) + 6/3125*(625*r^4 + 3000*r^3 + 10800*r^2 +
25920*r + 31104)*b*e^(-5/6*r) - 12/3125*(625*r^4 + 3000*r^3 +
10800*r^2 + 25920*r + 31104)*e^(-5/6*r) + 6/3125*(625*r^5 + 3750*r^4
+ 18000*r^3 + 64800*r^2 + 155520*r + 186624)*e^(-5/6*r)
r=0; I23=-12/125*(25*r^2 + 60*r + 72)*a*e^(-5/6*r) + 6/625*(125*r^3 + 450*r^2 + 1080*r + 1296)*a*e^(-5/6*r) - 12/625*(125*r^3 + 450*r^2 + 1080*r + 1296)*b*e^(-5/6*r) + 6/3125*(625*r^4 + 3000*r^3 + 10800*r^2 + 25920*r + 31104)*b*e^(-5/6*r) - 12/3125*(625*r^4 + 3000*r^3 + 10800*r^2 + 25920*r + 31104)*e^(-5/6*r) + 6/3125*(625*r^5 + 3750*r^4 + 18000*r^3 + 64800*r^2 + 155520*r + 186624)*e^(-5/6*r); I23 
        
3456/625*a + 108864/3125*b + 746496/3125
var('x'); psi0 = (1/pi)^(1/4)*exp(-x^2/2); psi1= ( 2^(1/2) / pi^(1/4) )*x*exp(-x^2/2); psi2=(2^(-1/2))*(1/pi^(1/4))*(2*x^2 -1)*exp(-x^2/2); v1=.1*x^3 +.1*x^4; E0=1/2 ; E1=3/2 ; E2=5/2; (integral(v1*psi0*psi2,x,-oo,oo))^2/(E0-E2) 
        
Traceback (click to the left of this block for traceback)
...
To reenable the Lisp debugger set *debugger-hook* to nil.
#(integral(v1*1*psi0,x,-10,10))^2/(E0-E2) var('x'); psi0 = (1/pi)^(1/4)*exp(-x^2/2);psi1= ( 2^(1/2) / pi^(1/4) )*x*exp(-x^2/2); psi2=(2^(-1/2))*(1/pi^(1/4))*(2*x^2 -1)*exp(-x^2/2); v1=.1*x^3 +.1*x^4; E0=1/2 ; E1=3/2 ; E2=5/2; integral(v1*psi0^2,x,-oo,oo) 
        
0.075
n(-1/8*((0.3*sqrt(pi)*e^50*erf(5*sqrt(2)) - 113.2*sqrt(2))*sqrt(2)*e^(-50) + (0.3*sqrt(pi)*e^50*erf(5*sqrt(2)) - 92.8*sqrt(2))*sqrt(2)*e^(-50))^2/sqrt(pi)) 
       
 
        
0.200000000000000
x = var('x') def u(n): fac = ((2*factorial(n))^(-1/2))*(2^(1/4)) return fac * hermite(n, sqrt(2*pi)*x) * exp(-pi*x^2) u_0(x) = u(0) u_2(x) = u(2) u_4(x) = u(4) graph1 = plot(u_0(x), -3, 3, rgbcolor=(1, 0, 0)) graph2 = plot(u_2(x), -3, 3, rgbcolor=(0, 1, 0)) graph3 = plot(u_4(x), -3, 3, rgbcolor=(0, 0, 1)) show(graph1 + graph2 + graph3) 
       
x,a,b=var('x,a,b') assume(a>0) assume(b>0) integral(b*x^2*exp(-a*x^2),x,0,oo) 
        
1/4*sqrt(pi)*b/a^(3/2)
f,z=var('f,z') f(z)=cos(z)^2*sin(z) integral(f(z),z,0,pi) 
        
2/3
r=var('r') psi(r)=(1/(32*pi)^(1/2))*(2-r)*exp(-r/2) integral(r*psi(r)^2*4*pi*r^2,r,0,oo) 
        
6
r,theta =var('x,theta');R110(r)= ;P(theta)= ; int1=integral(r*R110(r)^2*r^2,r,0,oo) ; int2=integral(P(theta)^2,theta,0,pi); prod=int1*int2*2*pi; prod 
        
81/2
x,y=var('x,y');f(x,y)=x*y; integral(f(x,y),x,0,1,y,0,1) 
        
line 4
    integral(f(x,y),x,_sage_const_0 ,_sage_const_1 ;y,_sage_const_0
,_sage_const_1 )
                                                   ^
SyntaxError: invalid syntax
#{1/( 64 π )1/2 } r exp(-r/2) sin(θ) exp(+ i φ ) r,theta=var('r,theta') Rad(r)=(1/(64*pi)^(1/2))*r*exp(-r/2) P(theta)=sin(theta) integral(r*Rad(r)^2*r^2,r,0,oo)*integral(P(theta)^2*sin(theta),theta,0,pi)*2*pi 
        
5
var('Z'); assume(r1>0) ;assume(r2>0);assume(Z>0); phi2(r2)=(Z^3/pi)^(1/2)*exp(-Z*r2); phi1(r1)=(Z^3/pi)^(1/2)*exp(-Z*r1); va=(1/r1)*integral(4*pi*r2^2*phi2(r2)^2,r2,0,r1); vb=integral(4*pi*r2*phi2(r2)^2,r2,r1,oo); J=integral(4*pi*r1^2*(va+vb)*phi1(r1)^2,r1,0,oo); 
        
Traceback (click to the left of this block for traceback)
...
Is  r1  positive, negative, or zero?
va=x ; vb=x; J=integral(va+vb,x,0,2);J 
        
4