Chapter 10 - Position and Momentum

We can start using sympy to handle symbolic math (integrals and other calculus):

from sympy import *

init_printing(use_unicode=True)

x, y, z = symbols('x y z', real=True)
a, c = symbols('a c', nonzero=True, real=True)

integrate?

There are two ways to use the integrate function. In one line, like integrate(x,(x,0,1)) or by naming an expression and then integrating it over a range:

A = (c*cos((pi*x)/(2.0*a)))**2
A.integrate((x,-a,a),conds='none')

We’ll use both, at different times. For longer expressions, the second form can be easier to read and write.

First, just try the following, then we’ll re-create some examples in the book.

integrate(x,(x,0,1))

integrate(x**2,(x,0,1))

The cell below will return an odd set of conditions on the result. This is because the solver doesn’t want to assume anything about a and there is a special case where the answer would be different. If you look closely though, that special case isn’t physically realistic so to igore these special conditions, we add conds='none'. The next cell down does what you’d expect. From here on out, just add this to the integrate function and we’ll get what we expect.

A = (c*cos((pi*x)/(2.0*a)))**2
A.integrate((x,-a,a))

A = (c*cos((pi*x)/(2.0*a)))**2
A.integrate((x,-a,a), conds='none')

So this tells us the normalization constant should be $c=\frac{1}{\sqrt{a}}$. Check that it is normalized if we do that:

psi = 1/sqrt(a)*cos((pi*x)/(2.0*a))  # notice we can name the expression something useful.
B = psi**2
B.integrate( (x,-a,a), conds='none')

Because psi is a real function, we can calculate expectation values by integrating over $x$ or $x^2$ with psi**2:

C = x*psi**2
C.integrate( (x,-a,a), conds='none')

D = x**2 * psi**2
E = D.integrate( (x,-a,a), conds='none')

E.n()  # the .n() method approximates the numerical part. You can look at the full expression below.

E

Example 10.2¶

h = Symbol('hbar', real=True)

Use the diff function to take a derivative of a symbolic expression. For example:

diff(x**2, x)

# Solution goes here

# Solution goes here

Example 10.3¶

p = Symbol('p', real=True)

# Solution goes here

# Solution goes here