* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Quantum Mechanics - s3.amazonaws.com
Electron configuration wikipedia , lookup
Aharonov–Bohm effect wikipedia , lookup
Ensemble interpretation wikipedia , lookup
Atomic theory wikipedia , lookup
Quantum computing wikipedia , lookup
Atomic orbital wikipedia , lookup
Identical particles wikipedia , lookup
Quantum entanglement wikipedia , lookup
Quantum machine learning wikipedia , lookup
Quantum group wikipedia , lookup
Wave function wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Quantum key distribution wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Renormalization group wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Bell's theorem wikipedia , lookup
Quantum teleportation wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Coherent states wikipedia , lookup
History of quantum field theory wikipedia , lookup
Path integral formulation wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Hydrogen atom wikipedia , lookup
Double-slit experiment wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Renormalization wikipedia , lookup
Canonical quantization wikipedia , lookup
Quantum state wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Wave–particle duality wikipedia , lookup
EPR paradox wikipedia , lookup
Probability amplitude wikipedia , lookup
Particle in a box wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
School of Mathematical and Physical Sciences PHYS1220 PHYS1220 – Quantum Mechanics Lecture 6 August 29, 2002 Dr J. Quinton Office: PG 9 ph 49-21-7025 [email protected] 29 August, 2002 1 School of Mathematical and Physical Sciences PHYS1220 Simple Harmonic Oscillator Recall from mechanics Classically, U ( x) 1 2 1 kx m 2 x 2 2 2 The particle oscillates between x=A, where A – amplitude 1 1 The total energy of the system is Etot KE U kA2 m 2 A2 2 2 Any value of A (and hence Etot) is allowed Total Energy is zero if the particle is at rest at x=0 Quantum Mechanics T.I.S.E. U(x) d 2 ( x) 1 m 2 x 2 ( x) E ( x) 2 2m dx 2 2 d 2 ( x) 2mE m 2 ( x ) dx 2 2 Analysis complicated, so guess Gaussian function ( x) BeCx x 2 29 August, 2002 2 School of Mathematical and Physical Sciences PHYS1220 Simple Harmonic Oscillator II Substituting into T.I.S.E. the guess corresponds with the ground state 1 ( x) Be C m 2 E 1 2 classical m 2 x 2 This is only one solution. The excited states are polynomials multiplied by an exponential (Gaussian) function n ( x) f ( x)e Quantum Cx2 Penetration into barrier occurs Plot of probability shows large variation from classical predictions 29 August, 2002 3 School of Mathematical and Physical Sciences PHYS1220 Simple Harmonic Oscillator III Energy is quantised 1 En n , n 0,1, 2,3,... 2 U(x) n=0 is ground state, with zero-point energy 1 E0 2 E4 9 / 2 Result justifies Planck’s hypothesis regarding vibrational energies E3 7 / 2 E nhf , n 1, 2,3,... Correspondence principle applies because lim En En1 E2 5 / 2 E1 3 / 2 E E0 / 2 n x but is negligible compared to the actual energy 29 August, 2002 4 School of Mathematical and Physical Sciences PHYS1220 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? 29 August, 2002 5 School of Mathematical and Physical Sciences PHYS1220 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. 2 After the wall 0 Reflection and Transmission coefficients may be developed such that R+T=1. If T<<1 then: T e 2GL G 29 August, 2002 2m(U 0 E ) 2 6 School of Mathematical and Physical Sciences PHYS1220 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) 29 August, 2002 7 School of Mathematical and Physical Sciences PHYS1220 Example 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 U 0 E (70eV 50eV) 1.6x1019 J/eV 3.2x10-18 J 2(9.11x1031 kg x 3.2x1018 J 2GL 2 (1.0x109 m) 46 34 1.06x10 J.s T e 2GL e 46 1.1x1020 which is extremely small (b) for L=0.1nm, 2GL = 4.6 T e2GL e 4.6 0.010 so by decreasing the barrier width by a factor of 10, the probability of tunnelling has increased by 18 orders of magnitude 29 August, 2002 8 School of Mathematical and Physical Sciences PHYS1220 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 29 August, 2002 • • • • • Transistors Quantum dots Conducting polymers Quantum computing Scanning Tunnelling Microscope 9 School of Mathematical and Physical Sciences PHYS1220 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 29 August, 2002 10 School of Mathematical and Physical Sciences PHYS1220 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 >1012 bytes of information. 29 August, 2002 STM image of Au on mica at T~70K 11 School of Mathematical and Physical Sciences PHYS1220 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 294 19 OCTOBER 2001 Writing with Atoms (achieved by pushing them around with the tip) 29 August, 2002 12 School of Mathematical and Physical Sciences PHYS1220 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’ and Bit states are controlled by light Registers are made from several finite well devices 29 August, 2002 13 School of Mathematical and Physical Sciences PHYS1220 Final Thought "Anyone who isn't shocked by quantum theory has not understood it." Neils Bohr, ~75 years ago 29 August, 2002 14