Postulates of quantum mechanics 373each of these results is obtained. This is precisely what we would expect the average
over many trials to yield as the number of trials goes to infinity, so that every possi-
ble outcome is ultimately obtained, including those with very low probabilities. The
quantity|αj|^2 gives the fraction of all the measurements that yieldajas a result.
We noted above that the act of measuring an operatorAˆcauses a “collapse” of
the state vector onto one of the eigenvectors ofAˆ. Given this, it follows that no
experiment can be designed that can measure two observables simultaneouslyunless
the two observables have a common set of eigenvectors. This is simply a consequence
of the fact that the state vector cannot simultaneously collapse onto two different
eigenvectors. Suppose two observables represented by Hermitian operatorsAˆandBˆ
have a common set of eigenvectors{|ak〉}so that the two eigenvalue equations
Aˆ|ak〉=ak|ak〉, Bˆ|ak〉=bk|ak〉 (9.2.29)are satisfied. It is then clear that
AˆBˆ|ak〉=akbk|ak〉
BˆAˆ|ak〉=bkak|ak〉
AˆBˆ|ak〉=BˆAˆ|ak〉
(AˆBˆ−BˆAˆ)|ak〉= 0. (9.2.30)Since|ak〉is not a null vector,AˆBˆ−BˆAˆmust vanish as an operator. The operator
AˆBˆ−BˆAˆ≡[A,ˆBˆ] (9.2.31)is known as thecommutatorbetweenAˆandBˆ. If the commutator between two opera-
tors vanishes, then the two operators have a common set of eigenvectors and hence can
be simultaneously measured. Conversely, two operatorsAˆandBˆthat do not commute
([A,ˆBˆ] 6 = 0) are said to beincompatible observablesand cannot be simultaneously
measured.
9.2.4 Time evolution of the state vector
So far, we have referred to the state vector|Ψ〉as a static object. In actuality, the
state vector is dynamic, and one of the postulates of quantum mechanics specifies how
the time evolution is determined. Suppose the system is characterized by a Hamilto-
nian operatorHˆ. The eigenvalues ofHˆ give the allowed energy levels of the system.
(How the Hamiltonian is obtained for a quantum mechanical system when the clas-
sical Hamiltonian is known will be described in the next subsection.) As inclassical
mechanics, the quantum Hamiltonian plays the special role of determining the time
evolution of the physical state. Quantum mechanics postulates that in the absence of
a measurement, the time dependence of the state vector|Ψ(t)〉is determined by
i ̄h∂
∂t|Ψ(t)〉=Hˆ|Ψ(t)〉, (9.2.32)which is known as thetime-dependent Schr ̈odinger equationafter the Austrian physi-
cist Erwin Schr ̈odinger (1887–1961) (for which he was awarded the Nobel Prize in