Schrodinger equation

• Basic of quantum mechanics from Wikipedia:
  • The quantum state of a system is a ray (equivalence class of vectors) in a separable (has a countable basis -> isomorphic to $\ell^2$) Hilbert space.
  • The Hilbert space associated with a composite system is the tensor product of spaces associated with the individual systems.
  • A physical observable is defined by a Hermitian matrix on H. Its expected value in a state represented by unit vector $\psi\in H$ is $\langle \psi | A | \psi \rangle$.
  • Somehow??? this implies a probability measure on values of A.
    • The Born rule. For any self-adjoint operator $A$ with discrete spectrum, the value measured from a normalized wavefunction is always one of the eigenvalues of $A$, $\lambda_i$, measured with probability $\langle \psi | \lambda_i \rangle\langle \lambda_i | \psi \rangle$ where $|\lambda_i \rangle$ is the corresponding eigenvector of $A$.
  • Technical point: the wavefunction is normalized in the sense that $\|\psi(t)\|_2 = \int |\psi(t)(x)|^2 dx = 1$. That is, at any given time $t$ we have a specific wave function \psi(t)(x), and its squared amplitude at any state $x$ gives a probability density. So: the wave function itself is a unit vector in the Hilbert space of wave functions.
• The Schrodinger picture of quantum mechanics says that state vectors $| \psi(t) \rangle$ evolve over time. The time evolution of the system is a function from real-valued $t$ to state vectors $|\psi(t)\rangle$. This function is defined by a differential equation, the Schrodinger equation:

where $\hbar$ is the Planck constant, and $H$ is a self-adjoint operator called the 'Hamiltonian' such that $\langle \psi | H | \psi \rangle$ is the total kinetic + potential energy of state $\psi$.

• If the system is stationary then we can use the time-independent form of the Schrodinger equation:

where $E$ is a constant energy level.