Dirac’s Postulates of Quantum Mechanics
I. Each pure state of a dynamical system is associated with a vector in a complex vector
space called “State Space” or “Ket Space”, K. The symbol for a ket vector corresponding
to a state A is |Ai.
II. Superposition Principle: The state |Ci is a superposition of states |Ai and |Bi if
and only if |Ci = a|Ai + b|Bi for complex numbers a, b.
III. The zero element of K corresponds to no state at all. A state superposed with itself is
either the same state or no state at all.
IV. There is a 1-1 correspondence between bras and kets, hA| ↔ |Ai such that
i)c∗ hA| ↔ c|Ai
ii)hA| + hB| ↔ |Ai + |Bi.
V. The bracket (inner product) hA|Bi satisfies
i)hA|Bi∗ = hB|Ai
ii)hA|Ai > 0 unless |Ai = 0 (Positive Definiteness).
Dirac’s postulates I-V are subsumed in Shankar’s Postulate I. Indeed Postulates II-V are
just the requirements that state space is a Hilbert space.
VI. The dynamical variables of a dynamical system correspond to linear operators on ket
space K. (Shankar Postulate II)
VII. If a system is in an eigenstate of a dynamical variable, an ideal measurement of that
dynamical variable yields the state’s eigenvalue with certainty. Conversely, if the measurement of a dynamical variable for some system yields a result with certainty, then the system
must be in an eigenstate of that dynamical variable belonging to the eigenvalue equal to that
result.
VIII. If a system is in a superposition of eigenstates of a dynamical variable belonging to
different eigenvalues, an ideal measurement of that dynamical variable yields one of those
eigenvalues and leaves the state in the corresponding eigenstate in the initial superposition.
We define an observable to be a dynamical variable whose eigenstates form a basis of K,
so it can be measured in any state.
IX. Let Ω be an observable and let the system be in state |ψi, with hψ|ψi finite. Then the
average of the results of a large number of measurements of Ω (with the system in |ψi for
each measurement) is given by hψ|Ω|ψi/hψ|ψi.
Dirac’s Postulates VII-IX are combined in Shankar’s Postulate III.
X. In Heisenberg Picture each dynamical variable A of a dynamical system is time dependent. If A has no explicit time dependence, the time evolution of A is given by a unitary
transformation U (t, t0 ):
A(t) = U † (t, t0 )A(t0 )U (t, t0 ).
Translated to Schrödinger Picture, the states evolve according to |ψ(t)i = U (t, t0 )|ψ(t0 )i.
(Shankar Postulate IV).
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c
2014
by Charles Thorn