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```{code-cell}
import sys
print(sys.executable)
```

```{code-cell}
:cell_style: center
:hide_input: false

import mmf_setup;mmf_setup.nbinit()
import logging;logging.getLogger('matplotlib').setLevel(logging.CRITICAL)
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt
```

(ass:1)=
# Assignment 1: Bras and Kets

Consider two bases: the standard basis $\{\ket{i}, \ket{j}\}$

## Polynomials

Here we consider the space of complex-valued functions on the interval $x\in [-1, 1]$,
for example.  To keep things finite, we will only consider polynomials of degree 3 or
less:

\begin{gather*}
  P(x) = a_3 x^3 + a_2 x^2 + a_1 x + a_0.
\end{gather*}

We can define an inner product using integration:

\begin{gather*}
  \braket{f, g} = \int_{-1}^{1}\d{x} f^*(x)g(x).
\end{gather*}

1. Show that this is a 4-dimensional Hilbert space.

Now consider the basis

\begin{gather*}
  \ket{x^n} \= x^n
\end{gather*}

2. Is this basis orthonormal?
3. Find an orthonormal basis $\{\ket{L_n}\}$ using the Gram-Schmidt procedure.  I.e.

   \begin{align*}
     \ket{L_0} &= \sqrt{\frac{1}{2}}\ket{x^0}, &
     \ket{L_1} &= \sqrt{\frac{3}{2}}\ket{x^1}, &
     &\cdots.
   \end{align*}

:::{margin}
These orthogonal polynomials $L_{m}(x)$ are the famous [Legendre polynomials] $P_m(x)$ up to an
overall normalization:

\begin{gather*}
  P_m(x) = \sqrt{m+\frac{1}{2}}L_m(x).
\end{gather*}
:::

Check your results against the following numerical solutions.  Note: the individual
basis functions $L_m(x)$ are unique up to a phase, so you may need to multiply your
solution by a phase like $-1$ to get it to agree with these numerical results prepared
using the numerical {ref}`sec:QR-Decomposition`

```{code-cell}
%matplotlib inline
import numpy as np, matplotlib.pyplot as plt

Nx = 100
Np = 5
x = np.linspace(-1, 1, 2*Nx+1)[1:-1:2]
dx = 2/Nx
n = np.arange(Np)
M = x[:, None]**n[None, :] * np.sqrt(dx)
Q, R = np.linalg.qr(M)
Lx = np.linalg.solve(R.T, M.T).T/np.sqrt(dx)
Lx /= np.sign(Lx[-1, :])[None, :]
assert np.allclose(Lx.T.conj() @ Lx * dx, np.eye(Np))

fig, ax = plt.subplots()
for m in range(Np):
    ax.plot(x, Lx[:, m], '-', label=f"$L_{{{m}}}(x)$")

Ps = [[1], [1, 0], [1.5, 0, -0.5], [2.5, 0, -1.5, 0]]
for m in range(len(Ps)):
    ax.plot(x, np.polyval(Ps[m], x)/np.sqrt(2/(2*m+1)), "--")
    
ax.set(xlabel="$x$", ylabel="$L_m(x)$");
plt.grid('on')
ax.legend(bbox_to_anchor=(1,1), loc="upper left");
```

[Legendre polynomials]: <https://en.wikipedia.org/wiki/Legendre_polynomials>