2ObjectivesIntroduce 5 postulates which relate to quantum mechanics.
3Outline The Physical Meaning Associated with the Wave Function Every Observable Has a Corresponding OperatorThe Result of an Individual MeasurementThe Expectation ValueThe Evolution in Time of a Quantum Mechanical System
414.1 The Physical Meaning Associated with the Wave Function Postulate 1The state of a quantum mechanical system is completely specified by a wave functionThe probability that a particle will be found at time t0 in a spatial interval of width dx centered at x0 is given by
514.1 The Physical Meaning Associated with the Wave Function For sound wave, the wave function is associated with the pressure at a time t and position x.For a water wave, is the height of the wave
614.1 The Physical Meaning Associated with the Wave Function The normalization condition for a particle confined in a 1-D space of infinite extent isΨ(x,t) must satisfy several mathematical conditions:Wave function must be a single-valued functionThe first derivative must be continuous functionWave function cannot infinite amplitude over a finite interval
714.2 Every Observable Has a Corresponding Operator Postulate 2For every measurable property of the system in classical mechanics such as position, momentum, andenergy, there exists a corresponding operator inquantum mechanics. An experiment in the laboratoryto measure a value for such an observable issimulated in the theory by operating on the wavefunction of the system with the correspondingoperator.
814.2 Every Observable Has a Corresponding Operator All quantum mechanical operators belong to a mathematical class called Hermitian operators that have real eigenvalues.
914.3 The Result of an Individual Measurement Postulate 3In any single measurement of the observable that corresponds to the operator , the only values that will ever be measured are the eigenvalues of that operator.
1014.3 The Result of an Individual Measurement The measured energy values of an atom are the eigenvalues of the time-independent Schrödinger equation:
1114.4 The Expectation Value Postulate 4 If the system is in a state described by the wave function , and the value of the observable a is measured once each on many identically prepared systems, the average value (also called the expectation value) of all of these measurements is given by
1214.4 The Expectation ValueAs eigenfunctions form an orthonormal set, it is normalized.Thus
1314.5 The Evolution in Time of a Quantum Mechanical System Postulate 5The evolution in time of a quantum mechanical system is governed by the time-dependent Schrödinger equation:
1414.5 The Evolution in Time of a Quantum Mechanical System We call this behavior deterministic in contrast to the probabilistic nature of Postulate 4.When time at t0, Postulate 4 applies.When t1 > t0, without carrying out a measurement in this time interval, Postulate 5 applies.If at time t1, we carry out a measurement again, Postulate 4 will apply.