# State Visualization

**State visualization** displays quantum states and operators for understanding and debugging. QuTiP provides plotting functions for populations, densities, Wigner functions, and more.

## Populations and Occupations

Plot populations (diagonal elements of density matrix):

```python
from qutip import *
import matplotlib.pyplot as plt

# Time evolution of a damped qubit
H = 0.5 * sigmaz()
c_ops = [0.1 * sigmam()]
times = np.linspace(0, 10, 100)
psi0 = basis(2, 1)

result = mesolve(H, psi0, times, c_ops, [sigmaz()])

# Plot expectation values
fig, ax = plt.subplots()
ax.plot(times, result.expect[0], 'b-', label='<σ_z>')
ax.set_xlabel('Time')
ax.set_ylabel('Expectation Value')
ax.legend()
plt.show()
```

## Density Matrix Heatmap

Visualize density matrix elements:

```python
rho = mesolve(H, psi0, [0, 5], c_ops, []).states[-1]

# Heatmap of |ρ_ij|
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4))
ax1.imshow(np.abs(rho.full()))
ax1.set_title('|ρ|')
ax2.imshow(np.angle(rho.full()))
ax2.set_title('arg(ρ)')
plt.show()
```

## Wigner Function

For harmonic oscillators, the Wigner function is a quasi-probability on phase space:

```python
# Coherent state
N = 20  # Hilbert space dimension
alpha = 2.0
psi = coherent(N, alpha)

# Wigner function
xvec = np.linspace(-4, 4, 200)
W = wigner(psi, xvec, xvec)

fig, ax = plt.subplots()
contourf = ax.contourf(xvec, xvec, W, levels=20)
ax.set_xlabel('Re(α)')
ax.set_ylabel('Im(α)')
plt.colorbar(contourf)
plt.show()
```

Visualization is essential for understanding quantum dynamics and debugging simulations.