Susskind Quantum Mechanics Common pitfalls

Common pitfalls

• Treating the quantum state like a classical state. A classical state lists actual properties. A quantum state gives the maximum information needed to compute probabilities for future measurements. See Quantum state.
• Confusing amplitudes with probabilities. Amplitudes are complex and can interfere. Probabilities are squared magnitudes. See Bra-ket notation.
• Forgetting the basis. A wavefunction or column vector is always the representation of a state in some chosen basis. Change the basis, and the components change. See Wavefunction.
• Assuming every observable has a pre-existing value. In the Susskind spin examples, a state sharp in σz is generally not sharp in σx. See Measurement and state preparation.
• Mixing up measurement and time evolution. Closed systems evolve unitarily. Measurements update the state relative to an observed outcome. Do not use one rule where the other belongs. See Quantum time evolution.
• Thinking noncommuting means “hard to measure.” The point is deeper: noncommuting observables do not share a full set of simultaneous eigenstates. See Commutators and compatible observables.
• Treating tensor products like ordinary multiplication. |a⟩ ⊗ |b⟩ is a state in a bigger vector space, not a number and not a state of either subsystem alone. See Tensor product states.
• Explaining entanglement as ignorance only. Classical correlation can come from ignorance about pre-existing facts. Entanglement is stronger: the whole can have a pure state while the parts do not. See Entanglement and Density matrix.
• Using entanglement as a faster-than-light telephone. Entanglement gives correlations after comparison; it does not let Alice choose Bob’s local outcome. See Susskind QM Lecture 7 - More on Entanglement.
• Taking “wavefunction” too literally. A wavefunction is a component list in a basis. It becomes a literal-looking function of position only after choosing the position basis. See Wavefunction.
• Dropping normalization. If the total probability is not one, the state-vector is not ready to predict measurement probabilities.
• Ignoring global phase. Multiplying a state by one overall phase does not change probabilities. Relative phases do matter.
• Expecting the oscillator ground state to have zero energy. The quantum harmonic oscillator has a nonzero zero-point energy. See Quantum harmonic oscillator.

Quick repair habit

When stuck, ask four questions: What is the state? What basis am I using? What observable is being measured? Is the system being measured or only evolving?