Superposition of eigenfunctions

Dept. of Math and Physics, Alfred State Coll. SUNY, Alfred NY 14802

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[Problem #8] A wavefunction at t=0 is shown in the figure: y(x,0)= k x for 0 < x < L and zero elsewhere.
a. Find the expression for k in terms of L.

b. Express it as a sum of eigenfunctions for one-dimensional infinite box yn(x,0). y(x,0)= a1*y1(x,0)+a2*y2(x,0)+... where yn(x,t)=sqrt(2/L)*sin(npx/L)*exp^-iWn/hbar*t, where Wn is the energy of the n-th eigenstate. Obtain the coefficint an.

c. Plot the sum y of the first 4 terms of the expansion.

[Solution]

restart:
y0:=k*x;
# The wavefunction at t=0.

# (a)

eq:=int(y0*y0,x=0..L)=1;
# Nornalization condition.

k:=solve(eq,k);
# Solve the equation above for k.

y0:=k[1]*x;
# MapleV returns two values for k. Assign the first one to k[1].

yn:=sqrt(2/L)*sin(n*Pi*x/L);
# Nornalized n-th eigenfunction for an infinitely deep well of width L.

# (b)

an:=int(yn*y0,x=0..L);
# The coefficints of the expansion.

an:=simplify(");
# Simplyfy the expression above.

y:=sum(an*yn,'n'=1..4);

# Add the first 4 terms of the expansion.
# y is the approximation to y(x,0), which is expanded in terms of the
# eigenfunctions for one-dimensional infine well: yn(x,0)=sqrt(2/L)*sin(npx/L)

L:=1;
# Set the length L=1.

#(c)

plot(y,x=0..L);
# Plot the graph of the synthesized function f.

L:=1;
# L=1 is used.

plot(y,x=0..L);
# Plot the synthesized function.

	You may want to change the number of terms n to add more terms. n=4 is used for the graph on the left.
	The graph on the left is obtained by adding n=100 terms. Note it is closer to the function being synthesized (the figure shown at the top.) Note also the sharp spike at L=1.

Created on 9/9/97 and last modified by T. Takeuchi on

takeuct@alfredstate.edu