The Schrödinger Wave Equation 2006 Quantum MechanicsProf. Y. F. Chen The Schrödinger Wave Equation

Classical Mechanics : Wave Mechanics = Geometrical optics: Wave optics classical mechanics Newtons theory = geometrical optics wave (quantum) mechanics Huygens theory = wave optics quantum phenomena diffraction & interference 2006 Quantum MechanicsProf. Y. F. Chen The Schrödinger Wave Eq. The Schrödinger Wave Equation classical mechanicsquantum mechanics

wave eq., solution is assumed to be sinusoidal, Helmholtz eq. de Broglie relation time-indep. Schrödinger eq. the appearance of Schrödinger imposed the quantum condition on the wave eq. of matter 2006 Quantum MechanicsProf. Y. F. Chen Time-independent Schrödinger Wave Eq. The Schrödinger Wave Equation Erwin Schrödinger

Einstein relation also represents the particle energy Schrödinger found a 1st-order derivative in time consistent with the time-indep. Schrödinger eq Quantum MechanicsProf. Y. F. Chen Time-dependent Schrödinger Wave Eq. The Schrödinger Wave Equation

the probability density of finding the particle wave function = field distribution its modulus square = probability density distribution the particle must be somewhere, total integrated = 1 (the wave function for the probability interpretation needs to be normalized. ) 2006 Quantum MechanicsProf. Y. F. Chen The Probability Interpretation The Schrödinger Wave Equation

N identically particles, all described by the number of particles found in the interval at t 2006 Quantum MechanicsProf. Y. F. Chen The Probability Interpretation The Schrödinger Wave Equation

a time variation of in a region is conserved by a net change in flux into the region. satisfy a continuity eq. by analogy with charge conservation in electrodynamics, conservation of probability 2006 Quantum MechanicsProf. Y. F. Chen The Probability Current Density The Schrödinger Wave Equation

is related to the phase gradient of the wave function., where = phase of the wave function t he larger varies with space, the greater 2006 Quantum MechanicsProf. Y. F. Chen Role of the Phase of the Wave Function The Schrödinger Wave Equation

reveals that the is irrotational only when has no any singularities, which are the points of. Conversely, the singularities of play a role of vortices to cause to be rotational Quantum MechanicsProf. Y. F. Chen Role of the Phase of the Wave Function The Schrödinger Wave Equation

with normalized is the probability of finding the momentum of the particle in in the neighborhood of p at time t 2006 Quantum MechanicsProf. Y. F. Chen Wave Functions in Coordinate and Momentum Spaces The Schrödinger Wave Equation

expectation value of r & p, find an expression for in coordinate space can be represented by the differential operator any function of p, & any function of r, can be given by 2006 Quantum MechanicsProf. Y. F. Chen Operators and Expectation values of Physical Variables The Schrödinger Wave Equation

CM all physical quantities can be expressed in terms of coordinates & momenta. QM all physical quantities can be given by any physical operator in quantum mechanics needs to a Hermitian operator Quantum MechanicsProf. Y. F. Chen Operators and Expectation values of Physical Variables The Schrödinger Wave Equation

the operators used in QM needs to be consistent with the requirement that their expectation values generally satisfy the laws of CM the time derivative of x can be given by integration by parts the classical relation between velocity and p holds for the expectation values of wave packets Quantum MechanicsProf. Y. F. Chen Time Evolution of Expectation values & Ehrenfests Theorem The Schrödinger Wave Equation

Ehrenfess theorem the time derivative of p can be given by has a form like Newtons 2nd law, written for expectation values for any operator A, the time derivative of can be given by (1) where is the Hamiltonian operator (2) the eq. is of the extreme importance for time evolution of expectation values in QM 2006 Quantum MechanicsProf. Y. F. Chen Time Evolution of Expectation values & Ehrenfests Theorem The Schrödinger Wave Equation

superposition of eigenstates based on the separation of t & r & (1) E = eigenvalue (2) = eigenfunction (3) stationary states if the initial state is represented by, independent of t 2006 Quantum MechanicsProf. Y. F. Chen Stationary States & General Solutions of the Schrödinger Eq. The Schrödinger Wave Equation