Quantum tunneling through an arbitrary potential barrier restart; Digits:=14: V:=exp(-5*x^2)+exp(-5*(x-2)^2);

LCYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiRJInhHRigiIiMhIiYiIiItRiQ2IywkKiQsJkYsRi8hIiNGL0YtRi5GLw==

plot(V,x=-2..5);

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

kE:=En->sqrt(2*En);

Zio2I0kjRW5HNiJGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JJXNxcnRHRiU2IywkOSQiIiNGJUYlRiU=

We need to propagate two independent solutions, and then to apply matching conditions. The Schroedinger wave equation is given as: SE:=En->-1/2*diff(u(x),x$2)+(V-En)*u(x)=0;

Zio2I0kjRW5HNiJGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS8sJi1JJWRpZmZHJSpwcm90ZWN0ZWRHNiQtSSJ1R0YlNiNJInhHRiUtSSIkR0YtNiRGMiIiIyMhIiJGNiomLCZJIlZHRiUiIiI5JEY4RjxGL0Y8RjwiIiFGJUYlRiU=

The boundary condition states that: 1) From the left we have an incident free plane wave with wavenumber k and of known amplitude, and a reflected free plane wave with unknown amplitude (whose magnitude squared will eventually become the reflection probability; (the choice of side from which the wave comes in is arbitrary). 2) To the right of the potential region we have an outgoing free plane wave with wavenumber k and unknown magnitude. The squared magnitude of the transmitted wave becomes the transmission probability (if the incident wave is normalized to unity). We know from (2) that the transmitted flux goes like Texp(i kx). Thus, te ratio of the derivative of the physical solution to the solution itself has to equal i k. Therefore, the strategy is to generate two independent solutions, and to form the linear combination that satisfies the requested relationship. The initial and final values for the independent varible range in the numerical integration of the differential equation are: x_i:=-2; x_f:=4;

ISIj

IiIl

We pick an energy value: En:=1/2;

IyIiIiIiIw==

We pick two intial conditions that provide independent solutions: IC1:=u(x_i)=1,D(u)(x_i)=1;

NiQvLUkidUc2IjYjISIjIiIiLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnRik=

IC2:=u(x_i)=1,D(u)(x_i)=0;

NiQvLUkidUc2IjYjISIjIiIiLy0tSSJERzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YmNiNGJUYnIiIh

sol1:=dsolve({SE(En),IC1},u(x),numeric,output=listprocedure): u1:=subs(sol1,u(x)): u1p:=subs(sol1,diff(u(x),x)): u1(x_f);

JCEzXz9jNCNIbEMrIiEjPA==

u1p(x_f);

JCIzYyRvWkJcYGhzKSEjPQ==

Now the second solution: sol2:=dsolve({SE(En),IC2},u(x),numeric,output=listprocedure);

NyUvSSJ4RzYiZio2I0YkNilJJV9yZXNHRiVJJV9kYXRHRiVJKl9zb2xucHJvY0dGJUkmX3hvdXRHRiVJJ19uZHNvbEdGJUkmX3BhcnNHRiVJI19pR0YlNiNJaW5Db3B5cmlnaHR+KGMpfjIwMDB+Ynl+V2F0ZXJsb29+TWFwbGV+SW5jLn5BbGx+cmlnaHRzfnJlc2VydmVkLkdGJUYlQytAJDIiIiI5I1lRQmludmFsaWR+aW5wdXQ6fnRvb35tYW55fmFyZ3VtZW50c0YlPkk4X0VudkRTTnVtZXJpY1NhdmVEaWdpdHNHRiVJJ0RpZ2l0c0dGJT5GOyIjOUAlL0ktX0VudkluRnNvbHZlR0YlSSV0cnVlRyUqcHJvdGVjdGVkRz44Jy0mSSZldmFsZkdGQjYjRjo2IzkkPkZELUZHRkk+OCUtSSVldmFsR0ZCNiMmSTxkc29sdmUvbnVtZXJpYy9kYXRhL21vZHVsZXNHSShfc3lzbGliR0YlNiMiIiM+OCYtX0ZOSSRHZXRHRiU2I1Evc29sbl9wcm9jZWR1cmVGJT44KS1JJG1hcEdGQjYkSSRyaHNHRkItRlo2I1EncGFyYW1zRiVAJDQtSSV0eXBlR0ZCNiRGRC5JKG51bWVyaWNHRkJDJEAwLUknbWVtYmVyR0ZCNiRGSjc4USZzdGFydEYlLkkmc3RhcnRHRiVRJ21ldGhvZEYlLkknbWV0aG9kR0YlUSVsZWZ0RiUuSSVsZWZ0R0YlUSZyaWdodEYlLkkmcmlnaHRHRiVRKWxlZnRkYXRhRiVRKnJpZ2h0ZGF0YUYlUStlbmdpbmVkYXRhRiVRKmV2ZW50c3RvcEYlLkkqZXZlbnRzdG9wR0YlUStldmVudGNsZWFyRiUuSStldmVudGNsZWFyR0YlUSxldmVudHN0YXR1c0YlLkksZXZlbnRzdGF0dXNHRiVRJ2xheHRvbEYlLkknbGF4dG9sR0YlUSdudW1mdW5GJS5JJ251bWZ1bkdGJUklTlVMTEdGQkMkPjgkLUZYNiMtSShjb252ZXJ0R0ZCNiRGSi5JJ3N0cmluZ0dGQkAoMkY1LUklbm9wc0dGQjYjNyNGX3JPRl9yLUZkbzYkRl9yLkkmYXJyYXlHRkJPLUZQNiRGX3JGNTBGX3JRKXByb2NuYW1lRiVGXXMtRltwNiRGSjcnUSVsYXN0RiUuSSVsYXN0R0YlUShpbml0aWFsRiUuSShpbml0aWFsR0YlRlxyQyRGXnJAJS1GZG82JEZfci5JJWxpc3RHRkJPJkZfcjYjRjVPRlxyLUZbcDYkRko3J1ErcGFyYW1ldGVyc0YlLkkrcGFyYW1ldGVyc0dGJVE3aW5pdGlhbF9hbmRfcGFyYW1ldGVyc0YlLkk3aW5pdGlhbF9hbmRfcGFyYW1ldGVyc0dGJUZcckMlPkZERmJyPkZfci1GWDYjRkRAJS9GREZddU83Iy1JJHNlcUdGQjYkLyZGaW42IzgqJkZfckZhdi9GYnY7RjUtRmpyNiNGaW5PNyRGZ3QtRl12NiQvRmB2JiZGXHNGVUZhdkZkdjMtRmRvNiRGREkiPUdGQi1GW3A2JC1JJGxoc0dGQkZndTcpRl10Rl50Rl11Rl51RmB1RmF1RlxyQyU+RkQvLUZjcjYkLUZmd0ZJRmVyLUZdb0ZJQCkvRmV3Rl10QCUtRmRvNiQtRl1vRmd1RmR0RmV1PkZfci1GWDYjL0ZddDclUSdzaW5nbGVGJUY1RmR4NEZieFlRZm5pbml0aWFsfmFuZC9vcn5wYXJhbWV0ZXJ+dmFsdWVzfm11c3R+YmV+c3BlY2lmaWVkfmlufmF+bGlzdEYlMy9GZXdGYHUvLUZqcjYjRmR4LCZGZnZGNUY1RjU+Rl9yLUZYNiMvRmV3NyVGanhGNS1JI29wR0ZCRmJ5RmV1QCdGYHhGZnQvRmV3Rl11Rmp1Rmh2M0Zgdy1GW3A2JEZldzcrUS1ldmVudGRpc2FibGVGJS5JLWV2ZW50ZGlzYWJsZUdGJVEsZXZlbnRlbmFibGVGJS5JLGV2ZW50ZW5hYmxlR0YlUStldmVudGZpcmVkRiUuSStldmVudGZpcmVkR0YlUSpkaXJlY3Rpb25GJS5JKmRpcmVjdGlvbkdGJUZcck8tRlg2I0Zqdy9GRFEuc29sbnByb2NlZHVyZUYlTy1GUDYjRlgvRkRRKHN5c3ZhcnNGJU8tRlo2I0ZmW2xAJTA5IUkodW5rbm93bkdGJU8tLkZcXGxGSUMlPjgoLUktdG9vbHMvZ2Vuc3ltR0ZUNiNRInhGJS1GUDYjLUkqRnJvbUluZXJ0R0ZCNiMtSTBfSW5lcnRfRlVOQ1RJT05HRiU2JC1JLF9JbmVydF9OQU1FR0YlNiNRJ2Fzc2lnbkYlLUkuX0luZXJ0X0VYUFNFUUdGJTYkLUkoVG9JbmVydEdGQjYjRmNcbC1JMF9JbmVydF9WRVJCQVRJTUdGJTYjLUkocG9pbnR0b0dGQjYjJi1GWjYjUTBzb2xuX3Byb2NlZHVyZXNGJUZodE8tRltdbDYjLUZeXWw2JEZnXWwtRmVdbDYjLUZoXWxGSVolQyRGZXVGZ3RGJVlGJUYlRiVGJS8tSSJ1R0YlRidmKkYnRihGMEYlQytGM0Y5RjxGPkZNRldGaG5AJEZib0MkQDBGam9GXXJGZ3NDJEZeckAlRmJ0TyZGX3JGVUZpdEZqdEMlRmR1RmV1QCVGaXVGanVPNyRGal9sRmp2Rl93QyVGaXdAKUZgeEAlRmJ4RmV1PkZfci1GWDYjL0ZddDclRmp4RlZGZHhGW3lGXHlGXnk+Rl9yLUZYNiMvRmV3NyVGanhGVkZpeUZldUAnRmB4RmlfbEZcekZqdUZdYGxGXXpGXVtsRmBbbEZiW2xGZVtsRmdbbEAlRltcbEZeXGxDJT5GY1xsLUZlXGw2I1EldSh4KUYlLUZQNiMtRltdbDYjLUZeXWw2JEZgXWwtRmVdbDYkRmddbC1GW15sNiMtRl5ebDYjJkZhXmxGVUZkXmxaJUMkRmV1RmpfbEYlRl5fbEYlRiVGJS8tSSVkaWZmR0ZCNiRGYF9sRiRmKkYnRihGMEYlQytGM0Y5RjxGPkZNRldGaG5AJEZib0MkQDBGam9GXXJGZ3NDJEZeckAlRmJ0TyZGX3I2IyIiJEZpdEZqdEMlRmR1RmV1QCVGaXVGanVPNyRGXmNsRmp2Rl93QyVGaXdAKUZgeEAlRmJ4RmV1PkZfci1GWDYjL0ZddDclRmp4RmBjbEZkeEZbeUZceUZeeT5GX3ItRlg2Iy9GZXc3JUZqeEZgY2xGaXlGZXVAJ0ZgeEZdY2xGXHpGanVGY2NsRl16Rl1bbEZgW2xGYltsRmVbbEZnW2xAJUZbXGxGXlxsQyU+RmNcbC1GZVxsNiNRLWRpZmYodSh4KSx4KUYlLUZQNiMtRltdbDYjLUZeXWw2JEZgXWwtRmVdbDYkRmddbC1GW15sNiMtRl5ebDYjJkZhXmxGX2NsRmRebFolQyRGZXVGXmNsRiVGXl9sRiVGJUYl

# not needed as we picked real-valued initial conditions #`dsolve/numeric/init_y0`:=subs(numeric=complex(numeric),eval(`dsolve/numeric/init_y0`)); sol2:=dsolve({SE(En),IC2},u(x),numeric,output=listprocedure): u2:=subs(sol2,u(x)): u2p:=subs(sol2,diff(u(x),x)): u2(x_f);

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u2p(x_f);

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We generated the two independent solutions using real boundary conditions. Now we assemble the complex-valued solution that satisfies the physical boundary condition of only a transmitted wave existing to the right of x_f. We need a single mixing coefficient, and determine it by imposing that the ratio of derivative over the solution equals i k. k:=kE(En);

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c2:='c2': c2:=solve((u1p(x_f)+c2*u2p(x_f))/(u1(x_f)+c2*u2(x_f))=I*k,c2);

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Now we can form the physical solution, which has arbitrary normalization at this point. We graph the real and imaginary parts: P1:=plot('Re(u1(x)+c2*u2(x))',x=-12..14,color=blue): P2:=plot('Im(u1(x)+c2*u2(x))',x=-12..14,color=red): plots[display](P1,P2);

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So far the incident wave has arbitrary normalization. Let us see whether we can figure the normalization out from the solution in region I (at x=x_i). We match now at the x_i a solution which has independent amplitudes for the incoming and reflected waves, named N and R respectively: eq1:=N*exp(I*k*x_i)+R*exp(-I*k*x_i)=u1(x_i)+c2*u2(x_i);

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eq2:=I*(k*N*exp(I*k*x_i)-k*R*exp(-I*k*x_i))=u1p(x_i)+c2*u2p(x_i);

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sol:=solve({eq1,eq2},{N,R});

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assign(sol); We observe that N and R are amplitudes, and turn out to be complex-valued. The properly normalized reflection coefficient now becomes: r:=R/N;

XiQkIS9ecUFGaUBlISM6JCEvKlFSJFJMaW0hIzk=

and the transmission coefficient: t:=evalf((u1(x_f)+c2*u2(x_f))/exp(I*k*x_f)/N);

XiQkIS9adSx0bHFPISM5JCEvLWBKJ1tgWSdGJQ==

The check whether this all makes sense is to sum the magnitude squared for reflection and transmission: abs(r)^2+abs(t)^2;

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Within numerical accuracy we preserve the norm; therefore we interpret as reflection and transmission probabilities: abs(r)^2,abs(t)^2;

NiQkIi9xJmVqZ0RaJSEjOSQiLyIzYXpYdV8mRiU=

Now we turn the calculation into a procedure: for given energy it calculates the reflection and transmission coefficients. Our previously developed commands are used step-by-step as before: Tunnel:=proc(En) local IC1,IC2,sol1,sol2,u1,u1p,u2,u2p,k,c2,N,R,r,t,sol,eq1,eq2; global V,kE,x_i,x_f; IC1:=u(x_i)=1,D(u)(x_i)=1; IC2:=u(x_i)=1,D(u)(x_i)=0; sol1:=dsolve({SE(En),IC1},u(x),numeric,output=listprocedure); u1:=subs(sol1,u(x)); u1p:=subs(sol1,diff(u(x),x)); sol2:=dsolve({SE(En),IC2},u(x),numeric,output=listprocedure); u2:=subs(sol2,u(x)); u2p:=subs(sol2,diff(u(x),x)); k:=kE(En); c2:=solve((u1p(x_f)+c2*u2p(x_f))/(u1(x_f)+c2*u2(x_f))=I*k,c2); N:='N': R:='R': eq1:=N*exp(I*k*x_i)+R*exp(-I*k*x_i)=u1(x_i)+c2*u2(x_i); eq2:=I*(k*N*exp(I*k*x_i)-k*R*exp(-I*k*x_i))=u1p(x_i)+c2*u2p(x_i); sol:=solve({eq1,eq2},{N,R}); assign(sol); r:=R/N; t:=evalf((u1(x_f)+c2*u2(x_f))/exp(I*k*x_f)/N); #print(" R= ",abs(r)^2," T= ",abs(t)^2," R+T= ",abs(r)^2+abs(t)^2," norm equal to 1.0 ?"); [abs(r)^2,abs(t)^2]; end: We check it for the previously tested energy value: Tunnel(1/2);

NyQkIi9xJmVqZ0RaJSEjOSQiLyIzYXpYdV8mRiU=

Now we are ready to scan the energy range from zero to above the peak height of the barrier. The results are accumulated in lists for subsequent plotting. Tr:=[]: Rf:=[]: for iE from 1 to 60 do: En:=iE/20; res:=Tunnel(En); Tr:=[op(Tr),[En,res[2]]]: Rf:=[op(Rf),[En,res[1]]]: od: P1:=plot(Rf,color=red): P2:=plot(Tr,color=blue): plots[display](P1,P2,labels=["E","{t,r}"]);

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

Observe the interesting features: 1) For sufficiently high energies only transmission occurs. Note, however, that classically this situation would be reached for E>1! 2) For E=0.66 perfect transmission and no reflection is observed. This is the phenomenon of a transition resonance. The resonance structure has a width associated with it. Most physical phenomena that have a lifetime associated with them (unstable elementary particles, alpha decay, nuclear gamma lines, atomic optical transitions, excitations in semiconductors, etc.) can be understood as processes where the wave equation describing the matter (or radiation) field involves a tunneling process. Broad resonances are associated with short lifetimes, sharp resonances with long lifetimes, according to Heisenberg's energy-time uncertainty relation. The transmission resonance is quite broad. We can make the walls of the two barriers thicker, and observe what the effect is on the transmission probability. We should adjust the right boundary: x_i:=-2; x_f:=5;

ISIj

IiIm

V:=exp(-8*x^6)+exp(-8*(x-3)^6);

LCYtSSRleHBHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2IywkKiRJInhHRigiIichIikiIiItRiQ2IywkKiQsJkYsRi8hIiRGL0YtRi5GLw==

plot(V,x=-2..5);

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

Tr:=[]: Rf:=[]: for iE from 1 to 240 do: En:=iE/80; res:=Tunnel(En); Tr:=[op(Tr),[En,res[2]]]: Rf:=[op(Rf),[En,res[1]]]: od: P1:=plot(Rf,color=red): P2:=plot(Tr,color=blue): plots[display](P1,P2,labels=["E","{t,r}"]);

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The transmission resonance is sharper now, and occurs just below the classical border value for the energy. Exercise: Observe what happens when the gap is widened between the potential humps. Be careful with the values of x_i, x_f, when changes are made to the potential. We can include an option in Tunnel to graph the solution. Perhaps this will help to understand why reflection and transmission comes about. Tunnel:=proc(En,iplot) local IC1,IC2,sol1,sol2,u1,u1p,u2,u2p,k,c2,N,R,r,t,sol,eq1,eq2; global V,kE,x_i,x_f,PL; IC1:=u(x_i)=1,D(u)(x_i)=1; IC2:=u(x_i)=1,D(u)(x_i)=0; sol1:=dsolve({SE(En),IC1},u(x),numeric,output=listprocedure); u1:=subs(sol1,u(x)); u1p:=subs(sol1,diff(u(x),x)); sol2:=dsolve({SE(En),IC2},u(x),numeric,output=listprocedure); u2:=subs(sol2,u(x)); u2p:=subs(sol2,diff(u(x),x)); k:=kE(En); c2:=solve((u1p(x_f)+c2*u2p(x_f))/(u1(x_f)+c2*u2(x_f))=I*k,c2); N:='N': R:='R': eq1:=N*exp(I*k*x_i)+R*exp(-I*k*x_i)=u1(x_i)+c2*u2(x_i); eq2:=I*(k*N*exp(I*k*x_i)-k*R*exp(-I*k*x_i))=u1p(x_i)+c2*u2p(x_i); sol:=solve({eq1,eq2},{N,R}); assign(sol); r:=R/N; t:=evalf((u1(x_f)+c2*u2(x_f))/exp(I*k*x_f)/N); if iplot=2 then print(" R= ",abs(r)^2," T= ",abs(t)^2," R+T= ",abs(r)^2+abs(t)^2," norm equal to 1.0 ?") fi; if iplot=1 then PL:=plot([V,'Re((u1(x)+c2*u2(x))/N)','Im((u1(x)+c2*u2(x))/N)'],x=x_i-10..x_f+10,color=[red,blue,green],numpoints=500,title=cat("Energy= ",convert(evalf(En,3),string))): fi; [abs(r)^2,abs(t)^2]; end: Tunnel(0.47,1);

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plots[display](PL);

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Tunnel(0.5,1);

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