The Schrödinger Wave Equation

The Schrödinger Wave Equation The Schrödinger Wave Equation 2006 Quantum Mechanics Prof. Y. F. Chen

The Schrödinger Wave Equation The Schrödinger Wave Eq. • Classical Mechanics : Wave Mechanics = Geometrical optics: Wave optics • classical mechanics ←→ Newton’s theory = geometrical optics wave (quantum) mechanics ←→ Huygen’s theory = wave optics • quantum phenomena ←→ diffraction & interference classical mechanics quantum mechanics 2006 Quantum Mechanics Prof. Y. F. Chen

The Schrödinger Wave Equation Time-independent Schrödinger Wave Eq. • 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 Erwin Schrödinger 2006 Quantum Mechanics Prof. Y. F. Chen

The Schrödinger Wave Equation Time-dependent Schrödinger Wave Eq. • Einstein relation： also represents the particle energy • → • Schrödinger found a 1st-order derivative in time consistent with the time-indep. Schrödinger eq. 2006 Quantum Mechanics Prof. Y. F. Chen

The Schrödinger Wave Equation The Probability Interpretation • 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 Mechanics Prof. Y. F. Chen

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

The Schrödinger Wave Equation The Probability Current Density • 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 Mechanics Prof. Y. F. Chen

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

The Schrödinger Wave Equation Role of the Phase of the Wave Function • 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. 2006 Quantum Mechanics Prof. Y. F. Chen

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

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

The Schrödinger Wave Equation Operators and Expectation values of Physical Variables • 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. 2006 Quantum Mechanics Prof. Y. F. Chen

The Schrödinger Wave Equation Time Evolution of Expectation values & Ehrenfest’s Theorem • 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. 2006 Quantum Mechanics Prof. Y. F. Chen

The Schrödinger Wave Equation Time Evolution of Expectation values & Ehrenfest’s Theorem • Ehrenfes’s theorem：the time derivative of p can be given by → has a form like Newton’s 2nd law, written for expectation values • for any operator A, the time derivative of <A> 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 Mechanics Prof. Y. F. Chen

The Schrödinger Wave Equation Stationary States & General Solutions of the Schrödinger Eq. • 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 Mechanics Prof. Y. F. Chen

The Schrödinger Wave Equation