2017-03-13-033549.sagews

# In this worksheet,you will calculate energies and wave functions of a particle in a one-dimensional box as a function of quantum number, n, and the length of the box, L.  The formula for calculating energy for a particle in a box is
var("n m h L E")

show(E==(n**2*h**2)/(8*m*L**2))

(n, m, h, L, E)

$\displaystyle E = \frac{h^{2} n^{2}}{8 \, L^{2} m}$

# where n is the quantum number,
# h is Planck's constant (Js),
# m is the mass of an electron (kg), and
# L is the length of the box (m)


E = ((n**2*h**2)/(8*m*L**2))
show(E)

$\displaystyle \frac{h^{2} n^{2}}{8 \, L^{2} m}$

h = 6.62606931*10**(-34)

m = 9.10938356*10**(-31)

# Quantum number, n, is an integer greater than or equal to 1.  Enter your value for n below:
n = 1

# L is the length of the box in meters. Insert your value for L in meters below:
L = 1*10**(-9)

E

1/8*h^2*n^2/(L^2*m)

# E.substitute(all of the values we assigned above) or simply copy and paste the equation and run
1/8*h^2*n^2/(L^2*m)

6.02466596830344e-20

# Now we will attempt to graph the energy as a function of n
# We can make a graph to show how energy changes as a function of n, leaving L constant
# Start by restarting the Sage worksheet. (Since we are ignoring the top of the worksheet, we have to redeclare our variables)
var("n m h L E")
E = n**2*h**2/(8*m*L**2)

(n, m, h, L, E)

h = 6.62606931*10**(-34)

m = 9.10938356*10**(-31)

# Insert your value for L in meters below:
L = 1*10**(-9)

#Let us take the time to now define E formally as a function of n
E(n) = n**2*h**2/(8*m*L**2)

E(1)
E(2)
E(3)
E(4)
E(5)
E(6)
E(7)
E(8)
E(9)
E(10)

6.02466596830344e-20 2.40986638732138e-19 5.42219937147310e-19 9.63946554928550e-19 1.50616649207586e-18 2.16887974858924e-18 2.95208632446869e-18 3.85578621971420e-18 4.87997943432579e-18 6.02466596830344e-18

plot(n**2*h**2/(8*m*L**2), xmin=0, xmax=10, color = "white") + point((1 , 6.02466596830344e-20), size= 50)+ point((2 , 2.40986638732138e-19), size= 50)+ point((3 , 5.42219937147310e-19), size= 50)+ point((4 , 9.63946554928550e-19), size= 50)+ point((5 , 1.50616649207586e-18), size= 50)+ point((6 , 2.16887974858924e-18), size= 50)+ point((7 , 2.95208632446869e-18), size= 50)+ point((8 , 3.85578621971420e-18), size= 50)+ point((9 , 4.87997943432579e-18), size= 50)+ point((10 , 6.02466596830344e-18), size= 50)

# While the above graph sucessfully shows E in terms of n, the coding is very cumbersome.  Fortunately I stumbled upon the command seen below which is much nicer. Thus, the finalized document will have cells 57-74 deleted.


list_plot([n**2*h**2/(8*m*L**2) for n in range(11)])

# Next we will try to graph energy as a function of n and L
# To begin this section, restart the cells once again and proceed from this point on.

var("n m h L E")
h = 6.62606931*10**(-34)
m = 9.10938356*10**(-31)
def E(n,L):
    return n**2*(6.62606931*10**(-34))**2/(8*(9.10938356*10**(-31))*L**2)

(n, m, h, L, E)

E(1, 1*10**(-9))

6.02466596830344e-20

 P = plot3d(E,(1,10), (1*10**(-9) , 1*10**(-6)))

P.show()

3D rendering not yet implemented

sqrt(4)

2

# Lastly, we are going to explore wave functions of different energy levels for a given box length L. The function f_n(x) is the wave function of energy level n.
# Restart now and continue

var("n m h L E")
h = 6.62606931*10**(-34)
m = 9.10938356*10**(-31)
L = 1 * 10**(-9)
def f(x,n):
    return sqrt(2/L)*sin(n*pi*x/L)

(n, m, h, L, E)

show(f(x,n))

$\displaystyle 20000 \, \sqrt{5} \sin\left(1000000000 \, \pi x\right)$

f(1,1)

0

n=1

plot(f(x,n), xmin=0, xmax=L)