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Chang-Kui Duan, Institute of Modern Physics, CUPT 1 1D systems Solve the TISE for various 1D potentials Free particle Infinite square well Finite square well Particle flux Potential step Transmission and reflection coefficients The barrier potential Quantum tunnelling Examples of tunnelling The harmonic oscillator

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Chang-Kui Duan, Institute of Modern Physics, CUPT 2 A Free Particle Free particle: no forces so potential energy independent of position (take as zero) Time-independent Schrödinger equation: Linear ODE with constant coefficients so try Combine with time dependence to get full wave function: General solution:

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Chang-Kui Duan, Institute of Modern Physics, CUPT 3 Notes Plane wave is a solution (just as well, since our plausibility argument for the Schr ö dinger equation was based on this assumption). Note signs in exponentials: –Sign of time term (-iωt) is fixed by sign adopted in time-dependent Schr ö dinger Equation –Sign of position term (±ikx) depends on propagation direction of wave. +ikx propagates towards +∞ while -ikx propagates towards –∞ There is no restriction on k and hence on the allowed energies. The states form a continuum.

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Chang-Kui Duan, Institute of Modern Physics, CUPT 4 Particle in a constant potential General solutions we will use over and over again Time-independent Schrödinger equation: Case 1: E > V (includes free particle with V = 0 and K = k) Case 2: E < V (classically particle can not be here) Solution:

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Chang-Kui Duan, Institute of Modern Physics, CUPT 5 Infinite Square Well Consider a particle confined to a finite length –a

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Chang-Kui Duan, Institute of Modern Physics, CUPT 6 Add and subtract these conditions: Even solution: ψ(x) = ψ(-x) Odd solution: ψ(x) = -ψ(-x) Infinite Square Well (2) Energy We have discrete states labelled by an integer quantum number

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Chang-Kui Duan, Institute of Modern Physics, CUPT 7 Infinite Square Well (3) Normalization Normalize the solutions Calculate the normalization integral Normalized solutions are

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Chang-Kui Duan, Institute of Modern Physics, CUPT 8 Sketch solutions Infinite Square Well (4) Wavefunctions Probability density Note: discontinuity of gradient of ψ at edge of well. OK because potential is infinite there. 3 1

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Chang-Kui Duan, Institute of Modern Physics, CUPT 9 Relation to classical probability distribution Infinite Square Well (5) Classically particle is equally likely to be anywhere in the box so the high energy quantum states are consistent with the classical result when we can’t resolve the rapid oscillations. This is an example of the CORRESPONDENCE PRINCIPLE. Quantum probability distribution is But

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Chang-Kui Duan, Institute of Modern Physics, CUPT 10 Energy can only have discrete values: there is no continuum of states anymore. The energy is said to be quantized. This is characteristic of bound- state problems in quantum mechanics, where a particle is localized in a finite region of space. The discrete energy states are associated with an integer quantum number. Energy of the lowest state (ground state) comes close to bounds set by the Uncertainty Principle: The stationary state wavefunctions are even or odd under reflection. This is generally true for potentials that are even under reflection. Even solutions are said to have even parity, and odd solutions have odd parity. Recover classical probability distribution at high energy by spatial averaging. Warning! Different books differ on definition of well. E.g. –B&M: well extends from x = -a/2 to x = +a/2. Our results can be adapted to this case easily (replace a with a/2). –May also have asymmetric well from x = 0 to x = a. Again can adapt our results here using appropriate transformations. Infinite Square Well (5) – notes

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Chang-Kui Duan, Institute of Modern Physics, CUPT 11 Finite Square Well Now make the potential well more realistic by making the barriers a finite height V 0 V(x) x -aa V0V0 IIIIII Region I:Region II:Region III: i.e. particle is bound

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Chang-Kui Duan, Institute of Modern Physics, CUPT 12 Finite Square Well (2) Boundary conditions: match value and derivative of wavefunction at region boundaries: Solve: Match ψ: Match dψ/dx: Now have five unknowns (including energy) and five equations (including normalization condition)

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Chang-Kui Duan, Institute of Modern Physics, CUPT 13 Finite Square Well (3) Cannot be solved algebraically. Solve graphically or on computer Even solutions when Odd solutions when We have changed the notation into q

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Chang-Kui Duan, Institute of Modern Physics, CUPT 14 Finite Square Well (4) Graphical solution Even solutions at intersections of blue and red curves (always at least one) Odd solutions at intersections of blue and green curves k 0 = 4 a = 1

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Chang-Kui Duan, Institute of Modern Physics, CUPT 15 Sketch solutions Finite Square Well (5) WavefunctionsProbability density Note: exponential decay of solutions outside well

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Chang-Kui Duan, Institute of Modern Physics, CUPT 16 Finite Square Well (6): Notes Tunnelling of particle into “forbidden” region where V 0 > E (particle cannot exist here classically). Amount of tunnelling depends exponentially on V 0 – E. Number of bound states depends on depth of well, but there is always at least one (even) state Potential is even, so wavefunctions must be even or odd Limit as V 0 →∞: We recover the infinite well solutions as we should.

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Chang-Kui Duan, Institute of Modern Physics, CUPT 17 Example: the quantum well Quantum well is a “sandwich” made of two different semiconductors in which the energy of the electrons is different, and whose atomic spacings are so similar that they can be grown together without an appreciable density of defects: Now used in many electronic devices (some transistors, diodes, solid-state lasers) Electron energy Position Material A (e.g. AlGaAs) Material B (e.g. GaAs)

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Chang-Kui Duan, Institute of Modern Physics, CUPT 18 Summary of Infinite and Finite Wells Infinite wellInfinitely many solutions Finite well Finite number of solutions At least one solution (even parity) Evanescent wave outside well. Even parity solutions Odd parity solutions Even parity Odd parity

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Chang-Kui Duan, Institute of Modern Physics, CUPT 19 Particle Flux In order to analyse problems involving scattering of free particles, need to understand normalization of free-particle plane-wave solutions. This problem is related to Uncertainty Principle: Momentum is completely defined Position completely undefined; single particle can be anywhere from -∞ to ∞, so probability of finding it in any finite region is zero Conclude that if we try to normalize so that we get A = 0. Solutions: Normalize in a finite box Use wavepackets (later) Use a flux interpretation

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Chang-Kui Duan, Institute of Modern Physics, CUPT 20 Particle Flux (2) More generally: what is the rate of change of probability that a particle is in some region (say, between x=a and x=b)? x a b Use time-dependent Schrödinger equation:

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Chang-Kui Duan, Institute of Modern Physics, CUPT 21 Particle Flux (3) x a b Interpretation: Flux entering at x=a Flux leaving at x=b minus Note: a wavefunction that is real carries no current

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Chang-Kui Duan, Institute of Modern Physics, CUPT 22 Particle Flux (4) Check: apply to free-particle plane wave. # particles passing x per unit time = # particles per unit length × velocity Makes sense: So plane wave wavefunction describes a “beam” of particles.

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Chang-Kui Duan, Institute of Modern Physics, CUPT 23 Particle Flux (5): Notes Particle flux is nonlinear Time-independent case: replace 3D case, Can use this argument to prove CONSERVATION OF PROBABILITY. Put a = -∞, b = ∞, then and

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Chang-Kui Duan, Institute of Modern Physics, CUPT 24 Potential Step Consider a potential which rises suddenly at x = 0: x Case 1: E > V 0 (above step) x < 0, V = 0 Boundary condition: particles only incident from left V(x) 0 V0V0 Case 1 x > 0, V =V0V0

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Chang-Kui Duan, Institute of Modern Physics, CUPT 25 Potential Step (2) Continuity of ψ at x = 0: Solve for reflection and transmission amplitudes:

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Chang-Kui Duan, Institute of Modern Physics, CUPT 26 Potential Step (3) Transmission and Reflection Fluxes x < 0x > 0 Check: conservation of particles Calculate transmitted and reflected fluxes (cf classical case: no reflected flux)

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Chang-Kui Duan, Institute of Modern Physics, CUPT 27 Potential Step (4) Case 2: E < V 0 (below step) Solution for x > 0 is now evanescent wave Matching boundary conditions: Transmission and reflection amplitudes: V(x) 0 V0V0 Solution for x < 0 same as before Transmission and reflection fluxes: This time we have total reflected flux.

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Chang-Kui Duan, Institute of Modern Physics, CUPT 28 Potential Step (5): Notes Some tunnelling of particles into classically forbidden region even for energies below step height (case 2, E < V 0 ). Tunnelling depth depends on energy difference But no transmitted particle flux, 100% reflection, like classical case. Relection probability is not zero for E > V 0 (case 1). Only tends to zero in high energy limit, E >> V (correspondence principle again).

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Chang-Kui Duan, Institute of Modern Physics, CUPT 29 Rectangular Potential Barrier Now consider a potential barrier of finite thickness: x b0 V(x) IIIIII V0V0 Boundary condition: particles only incident from left Region I: Region II:Region III: u = exp(ikx) + B exp(−ikx)u = C exp(Kx) + D exp(−Kx)u = F exp(ikx)

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Chang-Kui Duan, Institute of Modern Physics, CUPT 30 Rectangular Barrier (2) Match value and derivative of wavefunction at boundaries: Match ψ: Match dψ/dx: Eliminate wavefunction in central region: 1 + B = C + D 1 − B = K/(ik)(C − D) C exp(Kb) + D exp(−Kb) = F exp(ikb) C exp(Kb) − D exp(−Kb) = ik/K F exp(ikb)

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Chang-Kui Duan, Institute of Modern Physics, CUPT 31 Rectangular Barrier (3) Transmission and reflection amplitudes: For very thick or high barrier: Non-zero transmission (“tunnelling”) through classically forbidden barrier region. Exponentially sensitive to height and width of barrier. F |F| 2 =

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Chang-Kui Duan, Institute of Modern Physics, CUPT 32 Examples of Tunnelling Tunnelling occurs in many situations in physics and astronomy: 1. Nuclear fusion (in stars and fusion reactors) V Nuclear separation x Repulsive Coulomb interaction Incident particles Strong nuclear force (attractive) Assume a Boltzmann distribution for the KE, Probability of nuclei having MeV energy is Fusion (and life) occurs because nuclei tunnel through the barrier

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Chang-Kui Duan, Institute of Modern Physics, CUPT 33 Examples of Tunnelling 2. Alpha-decay Distance of α-particle from nucleus V Initial α-particle energy α-particle must overcome Coulomb repulsion barrier. Tunnelling rate depends sensitively on barrier width and height. Explains enormous range of α-decay rates, e.g. 232 Th, t 1/2 = yrs, 218 Th, t 1/2 = s. Difference of 24 orders of magnitude comes from factor of 2 change in α-particle energy!

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Chang-Kui Duan, Institute of Modern Physics, CUPT 34 Examples of Tunnelling 3. Scanning tunnelling microscope STM image of Iodine atoms on platinum. The yellow pocket is a missing Iodine atom A conducting probe with a very sharp tip is brought close to a metal. Electrons tunnel through the empty space to the tip. Tunnelling current is so sensitive to the metal/probe distance (barrier width) that even individual atoms can be mapped. If a changes by 0.01A (~1/100 th of the atomic size) then current changes by a factor of 0.98, i.e. a 2% change, which is detectable Tunnelling current proportional to and so Vacuum Material V x Probe a

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Chang-Kui Duan, Institute of Modern Physics, CUPT 35 Particles can tunnel through classically forbidden regions. Transmitted flux decreases exponentially with barrier height and width Summary of Flux and Tunnelling The particle flux density is We get transmission and reflection at potential steps. There is reflection even when E > V 0. Only recover classical limit for E >> V 0 (correspondence principle)

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Chang-Kui Duan, Institute of Modern Physics, CUPT 36 Simple Harmonic Oscillator Example: particle on a spring, Hooke’s law restoring force with spring constant k: Mass m x Time-independent Schrödinger equation: Problem: still a linear differential equation but coefficients are not constant. Simplify: change to dimensionless variable V(x) x

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Chang-Kui Duan, Institute of Modern Physics, CUPT 37 Simple Harmonic Oscillator (2) Asymptotic solution in the limit of very large y: Try it: Equation for H(y):

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Chang-Kui Duan, Institute of Modern Physics, CUPT 38 Simple Harmonic Oscillator (3) Solve this ODE by the power-series method (Frobenius method): Find that series for H(y) must terminate for a normalizable solution Can make this happen after n terms for either even or odd terms in series (but not both) by choosing Hence solutions are either even or odd functions (expected on parity considerations) Label normalizable functions H by the values of n (the quantum number) H n is known as the nth Hermite polynomial. 0

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Chang-Kui Duan, Institute of Modern Physics, CUPT 39 Simple Harmonic Oscillator (4) EXAMPLES OF HERMITE POLYNOMIALS AND SHO WAVEFUNCTIONS are normalization constants is a polynomial of degree

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Chang-Kui Duan, Institute of Modern Physics, CUPT 40 wavefunctionProbability density High n state (n=30) Decaying wavefunction tunnels into classically forbidden region Spatial average for high energy wavefunction gives classical result: another example of the CORRESPONDENCE PRINCIPLE Simple Harmonic Oscillator (5) Wavefunctions

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Chang-Kui Duan, Institute of Modern Physics, CUPT 41 Summary of Harmonic Oscillator 1) The quantum SHO has discrete energy levels because of the normalization requirement 2) There is ‘zero-point’ energy because of the uncertainty principle. 3) Eigenstates are Hermite polynomials times a Gaussian 4) Eigenstates have definite parity because V(x) = V(-x). They can tunnel into the classically forbidden region. 5) For large n (high energy) the quantum probability distribution tends to the classical result. Example of the correspondence principle. 6) Applies to any SHO, eg: molecular vibrations, vibrations in a solid (phonons), electromagnetic field modes (photons), etc

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Chang-Kui Duan, Institute of Modern Physics, CUPT 42 Example of SHOs in Atomic Physics: Bose-Einstein Condensation 87 Rb atoms are cooled to nanokelvin temperatures in a harmonic trap. de Broglie waves of atoms overlap and form a giant matter wave known as a BEC. All the atoms go into the ground state of the trap and there is only zero point energy (at T=0). This is a superfluid gas with macroscopic coherence and interference properties. Signature of BEC phase transition: The velocity distribution goes from classical Maxwell-Boltzmann form to the distribution of the quantum mechanical SHO ground state.

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Chang-Kui Duan, Institute of Modern Physics, CUPT 43 Example of SHOs: Molecular vibrations VIBRATIONAL SPECTRA OF MOLECULES Useful in chemical analysis and in astronomy (studies of atmospheres of cool stars and interstellar clouds). SHO very useful because any potential is approximately parabolic near a minimum V(x) Nuclear separation x H2 molecule SHO levels H H x

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