12.6 The Heisenberg Uncertainty Principle Flashcards
What does the Heisenberg uncertainty principle state?
The Heisenberg uncertainty principle states that it is not possible in principle to determine both the position and the momentum of an electron with infinite precision at the same time. The more precisely the position is known the more imprecisely the momentum is known and vice versa.
delta x times delta p >= h/4pi
delta x and delta p are the uncertainties in the measurements of position and momentum and h is the Planck constant.
How does the principle apply to measurements of energy and time?
In measuring the energy of a state a measurement that takes time delta t to complete results in an uncertainty delta E in the measured value of the energy such that:
delta E times delta t >= h/4pi
What is important about momentum being related to wavelength through the De Broglie wavelength?
A well defined wavelength implies a well defined momentum. In turn this implies a poorly defined position such that we don’t know where the electron is.
What can be worked out via the uncertainty principle?
The uncertainty principle may be used to makes estimates of various quantities.
What happens if there is a barrier and a proton wants to get through it according to classical theory?
Consider the wave function of a proton approaching an energy barrier from a region. The barrier consists of a positive potential. In classical theory to make it over the barrier and appear in the region past the barrier the protons total energy must be larger than eV where V is the size of the potential barrier. If it isn’t the proton will be reflected back into the region before the barrier, so the area past the barrier is forbidden to the proton.
What is tunnelling?
Something which makes it possible for the proton to get through the barrier into the region after the barrier. This is a consequence of the fact that particles have wave properties and are described by wave functions.
According to Schrodingers theory how does the tunnelling work?
Requires that the wave function of the proton in the regions before after and during the barrier join smoothly together. As a consequence the wave function cannot just stop to zero in the region after the barrier. It is as id the wave function leaks into this region. Notice that the De Broglie wavelength in the region after the barrier is the same as that in the region before the barrier, this means that the proton has the same energy after going through the barrier.
What is probability fo finding a particle in the region after the barrier?
The ratio of the square of the wave function in the region after the barrier to that in the region before the barrier. The there factors that affect this probability are the mass of the particle, the width of the barrier, the difference between the energy of the barrier and that of the particle. The larger each of these quantities is the smaller the transmission probability. So everything else being equal the transmission probability for an electron is greater than that for a proton.
What are the applications of tunnelling?
Scanning tunneling microscope (a microscope that can see atoms), tunnel diode (a diode in which the current can be very quickly stitched on and off).
