School of Mathematical and Physical Sciences PHYS August, PHYS1220 – Quantum Mechanics Lecture 6 August 29, 2002 Dr J. Quinton Office: PG 9 ph

School of Mathematical and Physical Sciences PHYS August, Simple Harmonic Oscillator Recall from mechanics Classically, The particle oscillates between x=  A, where A – amplitude The total energy of the system is Any value of A (and hence E tot ) is allowed Total Energy is zero if the particle is at rest at x=0 Quantum Mechanics T.I.S.E. Analysis complicated, so guess U(x) x Gaussian function

School of Mathematical and Physical Sciences PHYS August, Simple Harmonic Oscillator II Substituting into T.I.S.E. the guess corresponds with the ground state This is only one solution. The excited states are polynomials multiplied by an exponential (Gaussian) function Penetration into barrier occurs Plot of probability shows large variation from classical predictions classical Quantum

School of Mathematical and Physical Sciences PHYS August, Simple Harmonic Oscillator III Energy is quantised n=0 is ground state, with zero-point energy Result justifies Planck’s hypothesis regarding vibrational energies Correspondence principle applies because but is negligible compared to the actual energy U(x) x

School of Mathematical and Physical Sciences PHYS August, Barrier Potentials According to QM, wave functions can penetrate the walls of the potential (provided that U is finite) and there is a non-zero chance that the particle exists inside the wall region. Question: What if the wall is not infinitely thick?

School of Mathematical and Physical Sciences PHYS August, Tunnelling If we have a sufficiently energetic electron, and a thin wall or barrier, the electron may actually tunnel through the barrier. The solution to the bound particle in a finite well had the wavefunction decaying exponentially in the wall. If the wall is thin, there is a non-zero amplitude to the wavefunction at x=L. After the wall Reflection and Transmission coefficients may be developed such that R+T=1. If T<<1 then:

School of Mathematical and Physical Sciences PHYS August, Tunneling - II According to QM, the particle exists on both sides of the barrier. It just has a different probability of being on one side than the other It is not until you go to measure the particle (and collapse its wave function) that you actually know which side of the barrier it is on In going through the wall, no energy is lost (the particle’s energy is still E). Remember the amplitude is related to the probability, not the energy. The energy is related to the frequency (wavelength)

School of Mathematical and Physical Sciences PHYS August, Question: A 50 eV electron approaches a square barrier potential 70 eV high and (a) 1.0 nm thick (b) 0.10 nm thick. What is the probability that the electron will tunnel through? Answer: (a) First convert to SI units which is extremely small (b) for L=0.1nm, 2GL = 4.6 so by decreasing the barrier width by a factor of 10, the probability of tunnelling has increased by 18 orders of magnitude Example

School of Mathematical and Physical Sciences PHYS August, Applications of Quantum Mechanics ~ 30% of the US national Gross Domestic Product (GDP) is directly due to applications of Quantum Mechanics Semiconductor industry Digital communications with light/optic fibres Quantum mechanics has applications in many diverse areas, ranging from (but not limited to) Inert gas signs Semiconductors Lasers Microwave ovens MRI imaging Transistors Quantum dots Conducting polymers Quantum computing Scanning Tunnelling Microscope

School of Mathematical and Physical Sciences PHYS August, Inert Gas Signs Work on discharge lamp principle Gas is inert (or mixture) Ne, Ar, Kr Emission line spectrum has discrete wavelengths, usually one colour dominates (due to higher transition probability) By varying the pressure, it is possible to alter which electron states dominate, and hence alter the emission spectrum and therefore tune the sign’s colour to some degree

School of Mathematical and Physical Sciences PHYS August, Scanning tunnelling microscopy Vary the position of the tip above the surface to keep a constant tunnelling current and then plot the position control voltage against sample x,y dimension. Binning and Rohrer shared the 1986 Nobel prize. If this technology were used instead of optical techniques, a cd could store >10 12 bytes of information. STM image of Au on mica at T~70K

School of Mathematical and Physical Sciences PHYS August, Scanning Tunnelling Microscopy Reproducible Measurement of Single-Molecule Conductivity X. D. Cui, A. Primak, X. Zarate, J. Tomfohr, O. F. Sankey, A. L. Moore, T. A. Moore, D. Gust, G. Harris, S. M. Lindsay SCIENCE VOL OCTOBER 2001 Writing with Atoms (achieved by pushing them around with the tip)

School of Mathematical and Physical Sciences PHYS August, Quantum Computing Atomic scales are very small, device miniaturisation not an issue compared with current technology Quantum bits (qubits) are atomic or molecular spin states ‘up’ or ‘down’ Bit states are controlled by light Registers are made from several finite well devices

School of Mathematical and Physical Sciences PHYS August, Final Thought "Anyone who isn't shocked by quantum theory has not understood it." Neils Bohr, ~75 years ago