Susskind QM Lecture 9 - Particle Dynamics

One-line takeaway

The particle Schrödinger equation is the general time-evolution rule applied to the Hamiltonian of a particle.

Plain-language map

Lecture 9 asks how a particle moves once its state is represented by ψ(x,t). The abstract rule is already known:

iħ d|Ψ⟩/dt = H|Ψ⟩

To get wave mechanics, choose a Hamiltonian and represent it in the position basis.

Warm-up Hamiltonian

Susskind first uses a simple Hamiltonian proportional to momentum:

H = cP

In the position basis, this gives wave-packet solutions of the form ψ(x - ct). The packet moves at speed c while keeping its shape. This makes the link between the Hamiltonian and motion visible before adding more realistic dynamics.

Nonrelativistic particle

For an ordinary one-dimensional particle:

H = P²/(2m) + V(X)

In the position basis:

iħ ∂ψ/∂t = [-(ħ²/2m) ∂²/∂x² + V(x)] ψ

The kinetic term comes from the momentum operator squared. The potential term multiplies by V(x).

Core links

• Schrödinger equation
• Wavefunction
• Quantum time evolution
• Observables and eigenvalues

Common pitfalls

• Do not memorize the particle equation without remembering that it is a representation of the abstract equation.
• Do not forget that H is chosen from the physics of the system.
• Do not treat ψ(x,t) as a material fluid; it is an amplitude function whose squared magnitude gives probability density.

Links

• Susskind QM Lecture 8 - Particles and Waves
• Susskind QM Lecture 10 - Harmonic Oscillator
• Susskind Quantum Mechanics Examples