Postulates and Prediction

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Contents

Postulates, Informally Stated

  • State/Vector Representation
    • The state of a physical system as a whole is represented by the mathematical object |ψ⟩.
  • Observable/Operator Representation
    • Every measurable property of a physical system (an observable) is represented by a mathematical object called an Operator.
  • Time Evolution
    • As long as no observable of the system is measured, the value of |ψ⟩ evolves continuously and deterministically according to the Schrodinger Equation
  • Prediction
    • If the system is in state |ψ⟩ and an observable with operator M is measured, the probability that the observed value of M will be v is determined by |ψ⟩ and M.
  • Collapse
    • If an observable (with operator M) is measured with outcome v, the state of the system |ψ⟩ immediately after the measurement is determined by M and v (not by the Schrodinger Equation).

Prediction in Quantum Mechanics

  1. A physical system is defined.
  2. Certain observables of the system are measured at initial time T0, e.g. location, velocity, electric charge, energy, momentum.
  3. From the initial values of the observables and the Collapse Postulate, the initial state of the system (as a whole) |Ψ⟩ is calculated.  |Ψ⟩ is called the state vector of the system.  (Ψ is the Greek letter psi, pronounced sigh)
  4. From the initial value of |Ψ⟩ and the Time Evolution Postulate, the value of |Ψ⟩ at a future time Tf is calculated
  5. From the value of |Ψ⟩ at future time Tfand the Prediction Postulate, the probabilities of the possible values of the observables to be measured at time Tf are calculated.

Prediction in Classical versus Quantum Physics

Prediction in Classical Physics
  1. A physical system is defined.
  2. Certain observables (physical quantities) are measured at initial time T0, e.g. location, velocity, electric charge, energy, momentum.
  3. From the initial values of the observables and the postulates of Classical Physics, the values of the observables at any future time Tf. are calculated.
  4. The variables track the values of observables from T0 to Tf, e.g. the variable x tracks the location of a particular particle from T0 to Tf.
Prediction in Quantum Mechanics

Unlike Classical Physics, the variables of Quantum Mechanics do not track the values of observables between T0 and Tf. Quantum Mechanics, for example, predicts the probability that variable x, representing the location of a particle, has measured value v at future time TfBut it is silent on the value of x between T0 and Tf, when x is not measured.

Postulates, Precisely Stated

  • State/Vector Representation
    • The state of a physical system at a given time is represented by vector |ψ⟩ in a complex Hilbert space
  • Observable/Operator Representation
    • Every measurable property of a physical system (an observable) is represented by a Hermitian operator on the Hilbert space.
  • Time Evolution
    • As long as no observable of the system is measured, the state vector |ψ⟩ evolves according to the Schrodinger Equation, ℏ ∂|Ψ⟩/∂t = -iH|Ψ⟩, where H is the Hamiltonian of the system
  • Prediction
    • If the system is in state |ψ⟩ and an observable with operator M is measured, the probability of obtaining the outcome λi is P(λi) = |⟨Ψ|λi⟩|2, where λi is an eigenvalue of M and |λi⟩ is the corresponding eigenvector.
  • Collapse
    • If an observable (with operator M) is measured with outcome λ, the state vector |ψ⟩ of the system immediately after the measurement is an eigenvector of M with eigenvalue λ.