Modern Physics
3rd Edition
ISBN: 9781111794378
Author: Raymond A. Serway, Clement J. Moses, Curt A. Moyer
Publisher: Cengage Learning
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Chapter 7, Problem 3Q
To determine
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The wavefunction for a quantum particle tunnelling through a potential barrier of thickness L has the form ψ(x) = Ae−Cx in the classically forbidden region where A is a constant and C is given by C^2 = 2m(U − E) /h_bar^2 .
(a) Show that this wavefunction is a solution to Schrodinger’s Equation.
(b) Why is the probability of tunneling through the barrier proportional to e ^−2CL?
Consider the notes that can be played on a piano. In what way is a piano anexample of a quantized system? In this analogy, would a violin be continuousor quantized?
Calculate the transmission and reflection coefficients for an electron withkinetic energy 8 eV and potential energy 0 eV that moves into a regionwith potential energy of 5 eV.
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- What decreases the tunneling probability most: doubling the barrier width or halving the kinetic energy of the incident particle?arrow_forwardA 12.0-eV electron encounters a barrier of height 15.0 eV. If the probability of the electron tunneling through the barrier is 2.5 %, find its width.arrow_forwardWhich one of the following functions, and why, qualifies to be a wave function of a particle that can move along the entire real axis? (x)=Aex2; (x)=Aex; (x)=Atanx; (x)=A(sinx)/x; (x)=Ae|x|arrow_forward
- Can we simultaneously measure position and energy of a quantum oscillator? Why? Why not?arrow_forwardA simple model of a radioactive nuclear decay assumes that a-particles are trapped inside a well of nuclear potential that walls are the barriers of a finite width 2.0 fm and height 30.0 MeV. Find the tunneling probability across the potential barrier of the wall for a-particles having kinetic energy (a) 29.0 MeV and (b) 20.0 MeV. The mass of the a -particle is m=6.641027kg.arrow_forwardSuppose an infinite square well extends from L/2 to +L/2 . Solve the time-independent Schrödinger's equation to find the allowed energies and stationary states of a particle with mass m that is confined to this well. Then show that these solutions can be obtained by making the coordinate transformation x=xL/2 for the solutions obtained for the well extending between 0 and L.arrow_forward
- Consider a potential energy barrierbut whose height Ub is 6.0 eV and whose thickness L is 0.70 nm.What is the energy of an incident electron whose transmissioncoefficient is 0.0010?arrow_forwardConsider an electron of energy 1 eV that encounters a potential barrier of width 0.1nm and energy height 2ev. what is the probability of the electron crossing the barrier? repeat the same calculation for a protonarrow_forwardAn electron having total energy E = 4.50 eV approaches a rectangular energy barrier with U = 5.00 eV and L = 950 pm as shown. Classically, the electron cannot pass through the barrier because E < U. Quantum- mechanically, however, the probability of tunneling is not zero. (a) Calculate this probability, which is the transmission coefficient. (b) To what value would the width L of the potential barrier have to be increased for the chance of an incident 4.50-eV electron tunneling through the barrier to be one in one million?arrow_forward
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