Back to Quantum Mechanics

**Contents**

- Postulates, Informally Stated
- Prediction in Quantum Mechanics
- Prediction in Classical versus Quantum Physics
- Postulates, Precisely Stated

**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

- A physical system is defined.
- Certain observables of the system are measured at initial time T
_{0}, e.g. location, velocity, electric charge, energy, momentum. - 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*) - From the initial value of |Ψ⟩ and the Time Evolution Postulate, the value of |Ψ⟩ at a future time T
_{f}is calculated - From the value of |Ψ⟩ at future time T
_{f}and the Prediction Postulate, the probabilities of the possible values of the observables to be measured at time T_{f}are calculated.

**Prediction in Classical versus Quantum Physics**

**Prediction in Classical versus Quantum Physics**

**Prediction in Classical Physics**

- A physical system is defined.
- Certain observables (physical quantities) are measured at initial time T
_{0}, e.g. location, velocity, electric charge, energy, momentum. - From the initial values of the observables and the postulates of Classical Physics, the values of the observables at any future time T
_{f}. are calculated. - The
**variables track the values of observables**from T_{0}to T_{f}, e.g. the variable**x**tracks the location of a particular particle from T_{0}to T_{f}.

**Prediction in Quantum Mechanics**

**Prediction in Quantum Mechanics**

Unlike Classical Physics, the variables of Quantum Mechanics **do not track the values of observables **between T_{0} and T_{f}. Quantum Mechanics, for example, predicts the probability that variable **x**, representing the location of a particle, has measured value **v** at future time T_{f}**. **But it is silent on the value of **x** between T_{0} and T_{f}, when **x** is not measured.

**Postulates**, Precisely Stated

**Postulates**- 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 = -i
**H**|Ψ⟩, where**H**is the Hamiltonian of the system

- As long as no observable of the system is measured, the state vector |ψ⟩ evolves according to the Schrodinger Equation, ℏ ∂|Ψ⟩/∂t = -i
- Prediction
- If the system is in state |ψ⟩ and an observable with operator M is measured, the probability of obtaining the outcome λ
_{i}is P(λ_{i}) = |⟨Ψ|λ_{i}⟩|^{2}, where λ_{i}is an eigenvalue of M and |λ_{i}⟩ is the corresponding eigenvector.

- If the system is in state |ψ⟩ and an observable with operator M is measured, the probability of obtaining the outcome λ
- 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.