One-Dimensional Scattering of Waves

1. One-Dimensional Scattering of Waves One-Dimensional Scattering of Waves 2006 Quantum Mechanics Prof. Y. F. Chen

2. One-Dimensional Scattering of Waves 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 Mechanics Prof. Y. F. Chen

3. One-Dimensional Scattering of Waves The Transfer Matrix Method • 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 Mechanics Prof. Y. F. Chen

4. One-Dimensional Scattering of Waves The Transfer Matrix Method • 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 Mechanics Prof. Y. F. Chen

5. One-Dimensional Scattering of Waves The Transfer Matrix Method • 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 Mechanics Prof. Y. F. Chen

6. One-Dimensional Scattering of Waves The Transfer Matrix Method • 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 Mechanics Prof. Y. F. Chen

7. One-Dimensional Scattering of Waves The Transfer Matrix Method • 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 Mechanics Prof. Y. F. Chen

8. One-Dimensional Scattering of Waves The Transfer Matrix Method • 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 Mechanics Prof. Y. F. Chen

9. One-Dimensional Scattering of Waves The Transfer Matrix Method • 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 Mechanics Prof. Y. F. Chen

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

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

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

13. One-Dimensional Scattering of Waves The Potential Barrier • 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 Mechanics Prof. Y. F. Chen

14. One-Dimensional Scattering of Waves The Potential Barrier • 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 Mechanics Prof. Y. F. Chen

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

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

17. One-Dimensional Scattering of Waves Scattering of a Wave Package State • 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 Mechanics Prof. Y. F. Chen

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