Answer Happy

A particle of mass m and energy E > 0 is incident on a Pöschl-Teller potential (see Fig. 2) defined by: Wx) sech?(kot), where to is a real constant and the hyperbolic secant function is sechs = 2/6e* + € . V(x) E> 0 + X 0 - tk/m Figure 2: A particle of energy E > 0 is incident on a Pöschl-Teller potential from the far left

2 The classical expression for the total energy of the particle is: Pa + V«). 2m (a) Using the procedure of canonical quantisation, obtain the time-dependent Schrödinger equation that describes the corresponding one-dimensional quantum mechanical system. Take care to show your intermediate steps. (b) The solution to the time-independent Schrödinger equation for the Pöschl-Teller potential is: (a) 24 ks)on ( -4 ** - 2mE tanh) 12 where is a complex constant and the hyperbolic tangent function is: tant = { In the regions X - to, the solutions can be written as: Ae or for x- -- v) Be for x-too. Obtain expressions for the amplitudes A and B. Write an expression for the transmission coefficient T' in terms of the amplitudes A and B (and any other relevant constants). Show that the transmission coefficient T is equal to unity for the Pöschl-Teller potential. Why is this potential called reflectionless? (c) (d)