Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004) R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006) R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974) Lecture 4

Topic Today TOPIC FOR TODAY Solution for Free Particle Wavefunction Scattering from a Step Function (E > 0) Scattering from a Step Function (E < 0)

Free Particle Waves The general free-particle wavefunction is of the form                                                                                   which as a complex function can be expanded in the form Either the real or imaginary part of this function could be appropriate for a given application. In general, one is interested in particles which are free within some kind of boundary, but have boundary conditions set by some kind of potential. The free particle wavefunction is associated with a precisely known momentum: Function is not normalizable. Free particle has no definite energy.

Free Particle

The Free Particle General equation for free particle: This function can be normalized (for appropriate Ф(k) ). It carries a range of k values. This is referred to as wave-packets, superposition of sinusoidal waves which leads to interference, which allows localization and normalization. To determine Ф(k) so as to match the initial wave function use the expression below: Inverse Fourier Transform Fourier Transform

Example A free particle, which is initially localized in the range –a < x < a is released at time t = 0. where A and a are positive real constants. Find Ψ(x,t) Solution: Normalize Ψ(x,0)

Example Agrees with uncertainty principle: If spread in position is small, spread in momentum is large and vice-versa.

Wave Packet A wave packet is a superposition of sinusoidal functions whose amplitude is modulated by Ф. It consists of “ripples” and “envelopes”. A wave packet. The “envelope travels at the group velocity; the ripple” travel at the phase velocity. Particle velocity is group velocity.

Scattering from a Step Function (E > V) k12 x < 0 k22

Scattering from a Step Function (E > V) Boundary Conditions: Time dependent wave function for particle

Scattering from a Step Function (E > V)

Scattering from a Step Function (E < V) x < 0 x < 0

Scattering from a Step Function (E < V) If A2 is not equal to zero the wave function will not blow-up at infinity.

Scattering from a Step Function (E < V) T=1-R=0, Agrees with classical result. Wave function is not zero at x > 0, does not agree with classical result. This indicate probability of finding particle in classically forbidden region x > 0 is not zero. Penetration into classically forbidden region is a purely quantum mechanical effect.

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