Quantum Effects Negligible in Diffraction, Tunneling & Zero-Point Oscillation

[neelakash]

Homework Statement

I am to show that quantum effects are negligible in:

(i) The diffraction of a tennis ball of mass m=0.1 kg moving at a speed of 0.5 m/s
by a window of size 1X1.5 m^2

(ii)The tunneling probability for a marble of mass m=5 g moving at a speed of 10 cm/s against a rigid obstacle of height H=5 cm and width w=1 cm

(iii)The amplitude of the zero point oscillation for a pendulum of length l=1m and mass 1 kg

Homework Equations

The Attempt at a Solution

I show the first by: finding the de Broglie wavelength and arguing that for diffraction effects to be observable, we must have the incident ray's wavelength to be comparable in magnitude of the slit width length.

I show the third by: the QM amplitude is given by

$$\Delta$$$$\ x$$=$$\sqrt [{h}{/2m\omega}]$$

where h is actually h bar.(LaTeX did not accept \hbar)

Clearly the ampitude tends to zero for classical model given...

I am not sure of the correct expression of zero point amplitude in QM harmonic Oscillator but I get it in

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc4.html#c1

Please let me know if it is correct...

 

[neelakash]

due to some technical problem,there has been some problem with this thread...And another thread of the same name is appearing.I do not know why it is,but ifg it is my fault,I am expressing regret to Administrator

 

[neelakash]

I show the first by: finding the de Broglie wavelength and arguing that for diffraction effects to be observable, we must have the incident ray's wavelength to be comparable in magnitude of the slit width length.

I show the third by: the QM amplitude is given by

$$\Delta$$$$\ x=$$$$\sqrt{\frac{\hbar}{2m\omega}}$$

where h is actually h bar.(LaTeX did not accept \hbar)

Clearly the ampitude tends to zero for classical model given...

I am not sure of the correct expression of zero point amplitude in QM harmonic Oscillator but I get it in

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc4.html#c1

Please let me know if it is correct...


I found a formula of Transmission coeff here

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.html

T~exp[-2kL] where k~$$\sqrt{U-E}/h$$

Then the exponential is so small that the probability tends to zero.