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# Schrödinger equation (QuTiP)
[[schrodinger-equation|Schrödinger equation]] implementation using QuTiP.

QuTiP's `sesolve()` (**s**chrödinger **e**quation **solve**r) integrates the equation directly in state-vector form. It takes the Hamiltonian `H`, an initial ket `psi0`, and a time array `tlist`, then returns a list of ket vectors $\lvert\psi(t)\rangle$ at each time step — the numerical solution to:

$$i\hbar\frac{\mathrm{d}}{\mathrm{d}t}\lvert\psi\rangle = H\lvert\psi\rangle$$

The `e_ops` argument is a list of operators whose expectation values $\langle O\rangle = \langle\psi(t)\lvert O\lvert\psi(t)\rangle$ are computed at every step and stored in `result.expect`.

```python
# run:  python3 schrodinger_qutip.py
# desc: evolve a qubit in |+> under H = (omega/2)*Z and track <X> precessing on the Bloch sphere

import numpy as np
import qutip as qt

# Initial state: |+> = (|0> + |1>) / sqrt(2)
psi0 = (qt.basis(2, 0) + qt.basis(2, 1)).unit()

# Hamiltonian: Larmor precession around z-axis at angular frequency omega
omega = 2 * np.pi  # one full Bloch-sphere rotation over t in [0, 1]
H = 0.5 * omega * qt.sigmaz()

tlist = np.linspace(0, 1.0, 300)

# Integrate i * d/dt |psi> = H|psi>
result = qt.sesolve(H, psi0, tlist, e_ops=[qt.sigmax(), qt.sigmaz()])

# <X> oscillates as cos(omega*t); <Z> stays 0 (precession in the xy-plane)
print(result.expect[0][0],  result.expect[0][-1])   # ~1.0  ~1.0 (full period)
print(result.expect[1][0],  result.expect[1][-1])   # ~0.0  ~0.0
```