One-Dimensional Scattering of Waves 2006 Quantum MechanicsProf. Y. F. Chen One-Dimensional Scattering of Waves

in this chapter we will explore the phenomena of lD scattering to show that transmission is possible even when the quantum particle has insufficient energy to surmount the barrier the transfer matrix method will be utilized to analyze the one- dimensional propagation of quantum waves 2006 Quantum MechanicsProf. Y. F. Chen One-Dimensional Scattering of Waves

consider a particle of energy E and mass m to be incident from the left on arbitrarily shaped, 1D, smooth & continuous potential Such a problem can be solved by (1) dividing the potential into a piecewise constant function (2) using the transfer matrix method to calculate the probability of the particle emerging on the right-hand side of the barrier 2006 Quantum MechanicsProf. Y. F. Chen The Transfer Matrix Method One-Dimensional Scattering of Waves

Figure 6.1 Sketch of the quantum scattering at the jth interface between 2 successive constant values of the piecewise potential & the wave propagating through the constant potential until reaching the next interface at a distance after crossing the jth interface 2006 Quantum MechanicsProf. Y. F. Chen The Transfer Matrix Method One-Dimensional Scattering of Waves

the dynamics of the quantum particle is described by the Schrödinger eq., which is given in the jth region by the general solutions where & correspond to waves traveling forward and backward in jth region, respectively 2006 Quantum MechanicsProf. Y. F. Chen The Transfer Matrix Method One-Dimensional Scattering of Waves

the relationship between the coefficients & are determined by applying the boundary conditions at the interface as a result, it can be found that & is referred to be the scattering matrix 2006 Quantum MechanicsProf. Y. F. Chen The Transfer Matrix Method One-Dimensional Scattering of Waves & &

we can find that propagation between potential steps separated by distance carries phase information only so that a propagation matrix is defined as the successive operation of the scattering & propagation matrices leads to 2006 Quantum MechanicsProf. Y. F. Chen The Transfer Matrix Method One-Dimensional Scattering of Waves

for the general case of N potential steps, the transfer matrix for each region can be multiplied out to obtain the total transfer matrix the quantum particle is introduced from the left, the initial condition is given by if no backward particle can be found on the right side of the total potential 2006 Quantum MechanicsProf. Y. F. Chen The Transfer Matrix Method One-Dimensional Scattering of Waves

as a consequence, the transmission & reflection coefficients are given by those can be used to calculate the transmission & reflection probability of a quantum particle through an arbitrary 1D potential 2006 Quantum MechanicsProf. Y. F. Chen The Transfer Matrix Method One-Dimensional Scattering of Waves &

consider a particle of energy E and mass m that are sent from the left on a potential barrier 2006 Quantum MechanicsProf. Y. F. Chen The Potential Barrier One-Dimensional Scattering of Waves Figure 6.2 Sketch of the quantum scattering of a 1D rectangular barrier of energy V B

With the total matrix Q is given by where & it simplified as 2006 Quantum MechanicsProf. Y. F. Chen The Potential Barrier One-Dimensional Scattering of Waves

transmission probability in the case in terms of energy E and potential 2006 Quantum MechanicsProf. Y. F. Chen The Potential Barrier One-Dimensional Scattering of Waves

transmission probability in the case occurs whenever with the condition corresponds to resonances in transmission that occur when quantum waves back-scattered from the step change in barrier potential at positions & interfere and exactly cancel each other, resulting in zero reflection from the potential barrier 2006 Quantum MechanicsProf. Y. F. Chen The Potential Barrier One-Dimensional Scattering of Waves

transmission probability in the case (1) when, the transmission probability T 1 the particles are nearly not affected by the barrier & have total transmission (2) in the limit case, we have 2006 Quantum MechanicsProf. Y. F. Chen The Potential Barrier One-Dimensional Scattering of Waves

transmission probability in the case the wave number becomes imaginary, with if 2006 Quantum MechanicsProf. Y. F. Chen The Potential Barrier One-Dimensional Scattering of Waves

transmission probability in the case 2006 Quantum MechanicsProf. Y. F. Chen The Potential Barrier One-Dimensional Scattering of Waves Figure 6.3 Transmission probability as a function of particle energy for and several widths

in terms of &, the total wave function can be given by where is the Heaviside unit step func.,, the matrix element & are determined from the efficient & can be found to be given by 2006 Quantum MechanicsProf. Y. F. Chen Scattering of a Wave Package State One-Dimensional Scattering of Waves

where and the identities & are used to express the equation in a general form 2006 Quantum MechanicsProf. Y. F. Chen Scattering of a Wave Package State One-Dimensional Scattering of Waves