Question

It is possible to construct oscillatory wave packets without using trigonometric functions.

Consider the function y(x) = (64x^6 - 240x^4 + 180x^2 - 15)*e^(-x^2). Wave packets using polynomials occur in quantum mechanics

as solutions to the simple harmonic oscillator and the hydrogen atom, as we discuss later in this test.

(a) Sketch this function in the region where it has reasonably large amplitude.

(b) What is the width of this wave packet? Make a rough estimate from your sketch.

(c) Estimate the average wavelength.

(d) Estimate the uncertainty in the wavelength.

Answer 1

(a)

(b)

From the plot, we see that the wavefunction is very close to 0 for the distances |x|>4. Hence the width of the wavepacket is twice 4, which is 8.

(c) the average wavelength can be estimated from the distance between two consecutive crests. One of the crests is at -0.62 and the other is at 0.62. Hence the wavelength is 2*0.62 = 1.24

(d)

If we look at the difference between the second crest and the first crest, we get the difference as 2.164 - 0.62 = 1.544. This is the wavelength near x=1.5. Hence the uncertainty in the wavelength is given by