Presentation on theme: "Infinite Potential Well … bottom line"— Presentation transcript:
1 Infinite Potential Well … bottom line axV()=ElectronEnergies are quantized, defined by one single quantum number,n = 1, 2, 3, 4 …
2 Tunneling“ … If an electron comes up a potential barrier greater than its energy … there is a finite probability that it will “pass” through the barrier…”ABCDE
3 … for an electron !We place an electron in region I… with energy E less than VO (E<VO)… what is the probability the electron will be in I … II … III ??IV(x)o=aHow do we calculate the probability ??… we need to solve Schrodinger’s equation … apply boundary conditions etc.
4 Tunneling We now need to apply BC’s at x=0 and x=a … V(x)o=a2nd order linear differential equation.… (E-V) negative for zone II, hence no waveWe now need to apply BC’s at x=0 and x=a …The properties of ψ require that it be continuous and single valued
5 TunnelingIV(x)o=aIncidetRflA12yIyITherefore … the solution suggests that the electron can be found beyond the barrier VO … EVEN THOUGH its energy E is less than VO!
6 TunnelingWhat are the important factors that influence the tunneling probability ??… the energy of the electron… the width and height of the barrierFor a wide or high barrier …IV(x)o=ayncidetRflA12
7 Application of Tunneling xV()MetalyScondumbI(a)MterilsufcPobSnITuneligcrtaysv(A)xÅ
8 The Potential “Box” ¥ z c V y b a x = If you confine an electron in a box … what would you expect the wave-function to be?Think of it as a combination of 3 one-dimensional infinite potential wells… and thereforethe general solution will have the form of:yzxacbV=
9 The Potential “Box”The solution to the electron in a “box” problem results in 3 quantum numbersA specific solution or eigenfunction i.e.is called a state …Note that the electron energy is quantized and depends on 3 quantum numbers
10 The H-Atom … An Overview Describe the H-atom …i.e. what does the nucleus look like?how much charge is there at the nucleus?i.e. Z=1how many electrons?The H-atom represents the simplest system we can use to have a look at a real quantum physics example
11 The H-Atom … Force & PEObviously the electron is being attracted to the nucleus because of the …… Coulombic attraction between two opposite charges!The force between two charges is:and the potential energy is given by:
12 The H-Atom … Spherical Coordinates Due to the spherical symmetry of the H-atom … it makes sense to work in the spherical coordinate system instead of the cartesian one … i.e. x, y, z r,θ,φxyzrqNuclesfP(,)+ZinV(r)4peoZ2=+
13 The H-Atom … Wavefunction No need to go through the solution in detail …… we do however need to understand the origin of certain parameters and functions!Obtaining the wavefunction for the H-atom electron can be done by solving …… in 3-dimensionsi.e. one would expect to get… 3 quantum numbers!And the general wave function looks like:
14 The H-Atom … Quantum Numbers Two functions … R a function of r and Y a function of θ and φThree quantum numbers! … n, l, mlThe spherical part i.e. R depends on n and l… while the angular or spherical one, Y depends on l and mln=1,2,3,4,…… is the Principal Quantum Numberl=0,1,2,3 ……(n-1) is the Orbital Angular Momentum Quantum Numberml=-l, -(l-1), -(l-2), ……-2, -1, 0, 1, 2, …+l is the Magnetic Quantum Number… for now MEMORIZE these!
16 Let’s check the validity of these Energy States 3,2,22,3,12,3,01,2,42,3,-12,0,11,1,01,2,34,1,21,0,0
17 Energy ! Electron energies depend on n only … given by: What does this energy represent ?… the energy required to remove the electron from the n=1 state (i.e. to free the electron)also known as the ionization energy …Electrons prefer to minimize their energy … therefore most likely to be found in n=1 state known as the ground state!The value of Psi can be substituted back in
19 An electron with velocity 2. 1E6 m/s strikes a H atom An electron with velocity 2.1E6 m/s strikes a H atom. Find the n th energy level the electron will excite to. Calculate the wavelength of the light as the electron returns to ground state.K.E. = 12.5eV;n = > 3; ΔE = eV;λ = nm
20 Energy ! ¥ K E l e c t r o n g y , = 1 5 I i z . 3 6 V G u d s a C m =K1536VGudsa248IizCmfx
21 Orbital Angular Momentum Just like energy (En) … angular momentum L is also quantized… by ‘l’… what happens when l=0 ?xzOrbitngelcoqLBay
22 Orbital Angular Momentum For l=2 … ml would be ……-2, -1, 0, 1, 212ml=zLh(+)BextrnaxzOrbitngelcoqLBay
23 Selection Rules …Electron has momentum … also photons have intrinsic momentumWhen photons are absorbed … in addition to the energy conservation momentum must also be conserved …Selection rules: Δl=±1 … Δml=0, ±1i.e. if electron is in ground state 1,0,0 … (n,l,ml)If enough energy is gained to move up to n=2 then what are l and ml?l …0, 1 … and ml … -1, 0, 1Therefore … n=2, l=1, and ml=-1,0,1
25 Selection Rule Example An electron in State (3,2,-2). What are the energy states in Shell N, this electron can jump to?
26 Spin … (intrinsic angular momentum) Spin: last quantum number required to fully describe an electron!The component of the spin along a magnetic field is also quantized (i.e. if B-field is in Z-direction)
31 Energy states for multi electron atom 1s23456pdfKLMNOEnergy depends on both n and l
32 Pauli Exclusion Principle & Hund’s Rule NO two electrons can have the same set of quantum numbers …i.e. if one electron in ψ1,0,0,1/2 then a second electron in the same system will have … ψ1,0,0,-1/2Electrons in the same n, l orbital “like” to have “parallel" spins …