The Postulates of Quantum Mechanics There are six p ostulates of quantum mechanics Postulate The state of a quantum mechanical system is completely sp ecied by the function rt that dep ends on the co ordinates of the particle r and the time t This function is called the wavefunction or state function and has the prop ertythat rtrtd is the probability that the particle lies in the volume element d lo cated at r and time t This is the probabalistic interpretation of the wavefunction As a result the wavefunction must satisfy the condition that nding the particle somewhere in space is and this gives us the normalisation condition Z rtrtd The other conditions on the wavefunction that arise from the probabilistic interpretation are that it must be singlevalued continuous and nite We normally write wavefunctions with a normalisation constant included Postulate To every observable in classical mechanics there corresp onds a linear Hermitian op erator in quantum mechanics This p ostulate comes from the observation that the exp ectation value of an op erator that corresp onds to an observable must b e real and therefore the op erator must b e Hermitian Some examples of Hermitian op erators are Observable Classical Symbol Quantum Op erator Op eration p osition r r multiply by r j k momentum p p ihi x y z h kinetic energy T T m x y z p otential energy V r V r multiply by V r h V r total energy E H m x y z angular momentum l l ihy z x x z y l ihz l x y y x z l ihx y l z z y x Postulate In any measurement of the observable asso ciated with op erator A the only values that will ever b e observed are the eigenvalues a that satisfy the eigenvalue equation Aa This is the p ostulate that the values of dynamical variables are quantized in quantum mechanics although it is p ossible to have a continuum of eigenvalues in the case of unbound states If the system is in an eigenstate of A with eigenvalue a then any measurement of the quantity A will always yield the value a Although measurementwillalways yield a value the initial state do es not have to b e an eigenstate of A An arbitrary state can b e expanded in the complete set of eigenvectors of A A a as i i i n X c i i i where n maygo to innity In this case measurementof A will yield one of the eigenvalues a but we i dont knowwhichone The probability of observing the eigenvalue a is given by the absolute value of the i square of the co ecient jc j The third p ostulate also implies that after the measurement of yields i some value a the wavefunction col lapses into the eigenstate that corresp onds to a If a is degenerate i i i i collapses onto the degenerate subspace Thus the act of measurement aects the state of the system and this has b een used in many elegant exp erimental explorations of quantum mechanics eg Bells theorem Postulate If a system is in a state describ ed by the normalised wavefunction then the average value of the observable corresp onding to A is given by Z A d A Postulate The wavefunction or state function of a system evolves in time according to the timedep endentSchrodinger equation H rtih t Postulate The total wavefunction must be antisymmetric with resp ect to the interchange of all co ordinates of one fermion with those of another Electronic spin must b e included in this set of co ordinates The Pauli exclusion principle is a direct result of this antisymmetry p ostulate .