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An Electron Trapped in A Potential Well obability densities for an infinite well 2 / 2 / , 0 ) ( L x L x U 2 / , 2 / , ) ( L x L x x U ) ( ) ( ) ( d ) ( d 2 2 2 2 x E x x U x x m Solve Schrödinger equation 2 / , 2 / ) ( ) ( d ) ( d 2 2 2 2 L x L x x E x x x m outside the well 2 / , 2 / 0 ) ( L x L x x ) ( x U x 2 / L o 2 / L

An Electron Trapped in A Potential Well Probability densities for an infinite well Solve Schrödinger equation outside the well

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  • An Electron Trapped in A Potential WellProbability densities for an infinite wellSolve Schrdinger equationoutside the well

  • inside the wellAn Electron Trapped in A Potential Well

  • An Electron Trapped in A Potential WellNormalization For odd wave functionFor even wave functionprobability

  • inside the wellAn Electron Trapped in A Potential Wellenergydistribution

  • Barrier TunnelingDividing the space into three parts: I) to the left of the barrier II) within the barrier; and III) to the right of the barrierThe conditions for wave functions at the boundary are continuity.

  • Barrier TunnelingTunneling current

  • Si(111) Surface

  • Tetracene/Ag(110)[001]

  • Wave (matter wave)Schrdingers EquationprobabilityFree ElectronsHydrogen atomde Broglie relation, de Broglie wavelength.Potential WellParticleThe Nature of MatterEnergy quantizationProbability Barrier TunnelingSTM

  • Chapter 48Atomic Structure

  • Schrdingers Equation=0.529Bohr radius

  • ground stateEnergy quantizationSchrdingers Equation

  • The Uncertainty PrincipleThe angular momentum-angle Uncertainty Relationship

  • Schrdingers EquationAngular Momentum of Electrons in AtomsAngular momentum quantum numberAngular momentum quantization

  • Using this labeling, we can express 1s for ground state (n=1, l=0). The first excited state has two designations: 2s (n=2, l=0) and 2p (n=2, l=1). Angular momentum quantum numberl=0, 1, 2, , n-1.Ex. n=1, l=0 for s staten=2, l=0, 1. for l=0, s state and l=1, p state

  • Space quantizationml=-l, -(l-1), -1, 0, 1, ,(l-1), lmagnetic quantum number

  • n Principle quantum numberl Angular momentum quantum numberml=-l, -(l-1), -1, 0, 1, ,(l-1), lml magnetic quantum number

  • (n=1, l=0)The Ground Stateml=0, is spherically symmetric.

  • The 2s State (n=2, l=0)ml=0, is spherically symmetric.

  • An Excited State Of The Hydrogen AtomThe 2p State (n=2, l=1)ml=-1, 0, +1 is not spherically symmetric.

  • Electron Spin Pauli pointed out the need for a 4th quantum number in 1924Spin quantum numberandl Angular momentum quantum numberSpin magnetic quantum numbers Spin momentum quantum number

  • The States of Atomic Hydrogen The assembly of all hydrogen-atom states with the same principal quantum number n are said to form a shell. The collection of all states with the same value of the orbital angular momentum quantum number l is called a subshell. For a certain angular momentum l, there are 2l+1 states, and consider spin there are 2(2l+1) states.

  • statesstatesstatesExamplestates

  • The X-Ray Spectrum of AtomsThe Characteristic X-ray Spectrum

  • Atomic Magnetism How to study the angular momentum properties of the atom ?L is the orbital angular momentum vector of the electron.Lz=ml, ml=-l, -(l-1), ...-1, 0, +1, ..., +l

  • Atomic MagnetismLz=ml, ml=-l, -(l-1), ...-1, 0, +1, ..., +lIt can be expressed by Bohr magnetron B We can express the magnetic dipole moment in terms of the Bohr magnetron

  • The energy of electron

  • That is, atoms with different values of ml have different energies in the field, which provides a way to determine their orbital angular momentum.Atomic Magnetism

  • The Stern-Gerlach Experiment

  • there is no semiclassical corresponding. The full quantum mechanics givesAtomic Magnetismspin angular momentumSodium atom (Z=11)The Zeeman effect

  • Inverted populationLasers and Laser LightStimulatedemissionEmission

  • Lasers and Laser LightFour properties1) Laser light is highly monochromatic.2) Laser light is highly coherent3) Laser light is highly directional4) Laser light can be sharply focused.

  • ExercisesP1099 15, 16, 17P1100 28, 30