restart; Digits:=14: # Schroedinger QM in 1d: a double-well potential. # hbar=m=1 units + one more freedom V:=a*x^4-b*x^2+c;

LCgqJkkiYUc2IiIiIkkieEdGJSIiJUYmKiZJImJHRiVGJkYnIiIjISIiSSJjR0YlRiY=

# choose c such that potential is a minimum at E=0 (classical reference point) a:=1/16; b:=1/2;

IyIiIiIjOw==

IyIiIiIiIw==

Vp:=diff(V,x);

LCYqJEkieEc2IiIiJCMiIiIiIiVGJCEiIg==

sol0:=solve(Vp,x);

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x0:=sol0[2];

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c:=solve(eval(V,x=x0),c);

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plot(V,x=-2*x0..2*x0,view=[-2*x0..2*x0,0..2]);

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

# Question: will the ground state energy be well below the barrier? Vb:=eval(V,x=0);

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# what controls this question? [Heisenberg, zero-point energy] # what could give us a clue? Vt:=taylor(V,x=x0);

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# if we are interested in a pair of deep wells, then the Harmonic Oscillator approximation will tell us what the zero point energy might be. # here omega=sqrt(2) [since V=0.5*w^2*x^2 is the standard form], and we do not expect a zero-point energy below Vb 0.5*omega vs Vb! taylor(V,x=x0,3);

KycsJkkieEc2IiIiIiEiI0YmRiYiIiMtSSJPRyUqcHJvdGVjdGVkRzYjRiYiIiQ=

Vt2:=convert(taylor(V,x=x0,3),polynom);

KiQsJkkieEc2IiIiIiEiI0YmIiIj

w:=sqrt(solve(Vt2=0.5*w2*(x-x0)^2,w2));

JCIvSlBpTkA5OSEjOA==

# so how do we find out what the energies are for the ground state, and the first excited state? # pick a trial energy in accord with the above estimate: Et:=0.5*w;

JCIvYic9InkxcnEhIzk=

SE:=-diff(psi(x),x$2)+(V-Et)*psi(x)=0;

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ICe:=psi(0)=1,D(psi)(0)=0;

NiQvLUkkcHNpRzYiNiMiIiEiIiIvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0YlRidGKA==

sol:=dsolve({SE,ICe},numeric,output=listprocedure): psi0:=eval(psi(x),sol): psi0(0.5);

JCIzamdeOUY5QU01ISM8

# seems to work plot(psi0(x),x=-3*x0..3*x0,view=[-2.5*x0..2.5*x0,-2..2],numpoints=1000);

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Q&false6"/%*convertedGQ&false6"/%+imselectedGQ&false6"/%,placeholderGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"Q!6"-%%FONTG6%%(DEFAULTG%(DEFAULTG"#5%+HORIZONTALG%+HORIZONTALG-%%ROOTG6'-%)BOUNDS_XG6#$"#]!""-%)BOUNDS_YG6#$"$?"!""-%-BOUNDS_WIDTHG6#$"&g-"!""-%.BOUNDS_HEIGHTG6#$"%]P!""-%)CHILDRENG6"

# this energy is too high. [why do we know this?] Et:=0.5;

JCIiJiEiIg==

SE:=-diff(psi(x),x$2)+(V-Et)*psi(x)=0;

LywmLUklZGlmZkclKnByb3RlY3RlZEc2JC1GJTYkLUkkcHNpRzYiNiNJInhHRixGLkYuISIiKiYsKCokRi4iIiUjIiIiIiM7KiRGLiIiIyNGL0Y4JCIiJkYvRjVGNUYqRjVGNSIiIQ==

ICe:=psi(0)=1,D(psi)(0)=0;

NiQvLUkkcHNpRzYiNiMiIiEiIiIvLS1JIkRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiY2I0YlRidGKA==

sol:=dsolve({SE,ICe},numeric,output=listprocedure): psi0:=eval(psi(x),sol): psi0(0.5);

JCIzSCZleVFLMzAxIiEjPA==

# seems to work plot(psi0(x),x=-3*x0..3*x0,view=[-2.5*x0..2.5*x0,-2..2],numpoints=1000);

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derGQ&false6"/%6selection-placeholderGQ&false6"/%,mathvariantGQ'italic6"Q!6"-%%FONTG6%%(DEFAULTG%(DEFAULTG"#5%+HORIZONTALG%+HORIZONTALG-%%ROOTG6'-%)BOUNDS_XG6#$"#]!""-%)BOUNDS_YG6#$"$?"!""-%-BOUNDS_WIDTHG6#$"&g-"!""-%.BOUNDS_HEIGHTG6#$"%]P!""-%)CHILDRENG6"

# this is too low. # So let us look at a way to generate an acceptable solution. # Define a large x value X:=4*x0;

IiIp

# Define a function that takes Et as an input value, sets up the Schroedinger solution, and returns the value of the magnitude squared at x=X. psiX:=proc(Et) local SE,ICe,sol,psi0; global V,X; SE:=-diff(psi(x),x$2)+(V-Et)*psi(x)=0; ICe:=psi(0)=1,D(psi)(0)=0; sol:=dsolve({SE,ICe},numeric,output=listprocedure): psi0:=eval(psi(x),sol): psi0(X)^2; end: psiX(0.6);

JCIvMmI3W1I7QiIjOg==

# Now generate a sequence of points where the trial energy is varied. PP:=seq([0.55+0.01*i,psiX(0.55+0.01*i)],i=1..199): plot([PP],style=point);

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

# we have the ground state close to E=0.7, and the first excited state close to E=2.2