Probability and Normalization Although we don t know exactly where the particle might be inside the box, we know that it has to be in the box. This means that, ψ ( x) dx = 1 (normalization condition) L nπ L C sin x dx = C = 1 L 0 So, the normalization condition fixes the final free constant C in the wavefunction, C = L. This then gives, ( ) sin nπ x ψ n x = (particle in a box) L L
Probability in Finding the Particle Notes: The positions for the particles are probabilistic. We just know that it has to be in the box but the exact location within the box is uncertain. n =19 Not all positions between x = 0 and L are equally likely. In CM, all positions are equally likely for the particle in the box. There are positions where the particle has zero probability to be found.
Time Dependence Note that with ψ (x) found, we can write down the full wavefunction for the time-dependent Schrodinger equation as: Ψ (,) xt =ψ () xe iet n n / recall E = ω = hf Note that the absolute value for nπ x Ψ n( xt, ) = sin e L L iet / e iet / is unity, i.e., iet / iet / iet / 0 e = e e = e = 1 so that Ψ n (x,t) = ψ (x) is independent of time and probability density in finding the particle in the box is also independent of time.
Finite Square-Well Potential In Newton s mechanics, if E < U 0, a particle will be trapped inside the well. In QM, such a trapped state is called a bound state. If E > U 0, then the particle is not bound. Square-well with finite height For the infinitely deep well (as in the particle in a box problem), all states are bound states. U( x) U, 0 elsewhere = 0, 0 x L For a finite square-well, there will typically be only a finite number of bound states.
Finite Square-Well Potential Similar to the particle in a box problem, U(x) = 0 inside the well, we have or, ψ dx ψ ( x) = Ae + Ae d ( x) me = k ψ ( x), where k = ikx 1 ikx ψ inside( x) = Acos kx + B sin kx and A, B are constants to be determined by boundary conditions and normalization. (inside the well) But for a finite square-well potential, the wavefunction is not identically zero outside the well. The Schrodinger equation is given by: d ψ ( x) dx ( ) 0 = m ( U 0 E ) ψ( x) = κ ψ( x), where κ = mu E (outside the well)
Finite Square-Well Potential Since U 0 > E, κ is real and the wavefunction outside the well is given in terms of exponentials instead of harmonic functions: κx ψ ( x) = Ce + De κx where C and D are constants to be determined by B.C. and normalization again. For this problem, there is a new type of B.C. at large distances from the origin: wavefunction must remain finite (not blowing up) at large x ψ ( ) κx and ( ) 0 x = Ce ψ x = De x< x> L κx
Finite Square-Well Potential For a physical quantum particle, both ψ ( x) and dψ ( x)/ dx must be continuous at x = 0 and x = L. Matching ψ x< 0 ( x), ψinside( x),and ψ x> L( x) at x = 0 and x = L will enforce a certain set of allowed functions to be fitted within the well and the bound state energy is correspondingly quantized.
Example: e in a Square-Well/Quantum Dot = 7.6eV = 3.6eV = 0.94eV What is the wavelength of light released if the electron was originally at the 1 st excited state (n=) and relaxed back to the ground state (n=1)? hc hf = = E E1 λ hc λ = = = 460nm E E ev ev 1 ( 140eV s) ( 3.6 0.94 )
Example: e in a Square-Well/Quantum Dot Application: Quantum dots are nanometer-sized particles of a semi-conductor (such as cadmium selenide or gallium arsenide). An electron within a quantum dot behaves much like a particle in a finite square well potential. When a quantum dot is illuminated by a ultraviolet light, the electron within the quantum dot can be excited to a higher energy state (let say, n=3) from ground state (n=1). When it relaxed back to the ground state thru the intermediate state (n=): [3 and 1] photons with lower energy (longer wavelengths in the visible range) can be observed (fluorescence)! D Qdot
Tunneling Through a Barrier Consider the following potential barrier: U( x) U, 0 0 x L = 0, elsewhere A quantum particle with mass m and energy E is traveling from the left to the right. Classical Expectation (with E < U 0 ): In the region x < 0, the particle is free but when it reaches x = 0, the particle will hit a wall since its E is less than the potential at x =0. It will be reflected back and it could not penetrate the barrier!
Tunneling Through a Barrier Quantum Picture: x < 0 and x > L (free space): The wavefunction for a free particle with definite E and P is sinusoidal, e ikx or e -ikx. 0 x L(inside the barrier): E < U 0 wavefunction is a decaying exponential e -κx. exponential function within barrier
Tunneling Through a Barrier If energy is sufficiently high (but still E < U 0 ) and the barrier is not too wide so that the exponential decay does not significantly diminish the amplitude of the incidence wave, then there is a non-zero probability that a quantum particle might penetrate the barrier. (reduced amplitude reduced probability but not zero probability!) The transmission probability T can be solved from the Schrodinger equation by enforcing the boundary conditions: ( ) mu 0 E κ L E E T = Ge, κ =, G = 16 1 U0 U0 (for E/U 0 small)
Application of Tunneling (STM) Scanning Tunneling Microscope (STM): The tunneling current detected will vary sensitively on the separation L of the surface gap and these variations can be used to map surface features. In a STM, an extremely sharp conducting needle is brought very close to a surface that one wants to image. When the needle is at a positive potential with respect to the surface, electrons from the surface can tunnel through the surface-potential-energy barrier.
The Harmonic Oscillator Classically, the harmonic oscillator can be envisioned as a mass m acted on by a conservative force: F = k' x(hooke s Law: mass on a spring). Its associated potential energy is the familiar: 1 U( x) = k' x where k is the spring constant. For a classical particle with energy E, the particle will oscillate sinusoidally about x = 0 with an amplitude A and angular frequency. ω = k' m
The Harmonic Oscillator The Harmonic Oscillator is important since it is a good approximation for ANY potential U( x) near the bottom of the well.
The Harmonic Oscillator For the Quantum analysis, we will use the same form of the potential energy for a quantum Harmonic Oscillator. d ψ x + = m dx ( ) 1 k ' xψ ( x ) Eψ ( x ) or d ψ ( x) m 1 = k ' x E ψ ( x ) dx Boundary condition consideration: U(x) increases without bound as x so that the wavefunction for particle with a given energy E must vanish at large x. The solutions for this ordinary differential equation with the boundary condition ψ ( x) 0 as x ± are called the Hermite functions: ψ ( x) = Ce And we have the following quantized energies: mk ' x 1 En = n+ ω, n= 0,1,, (ground state n=0)
The Harmonic Oscillator Hermite Functions 1 En = n+ ω, n= 0,1,, note: wavefunction penetration into classically forbidden regions. note: similar to previous examples, the lowest E state is not zero.
The H-atom In the Schrodinger equation, we have explicitly included the Coulomb potential energy term under which the electron interacts with the nucleus at the origin: 1 e Ur () =, 4πε r 0 r = x + y + z is the radius in spherical coordinates.
The H-atom e - does not exist in well-defined circular orbits around the nucleus as in the Bohr s model. e - in a H-atom should be envisioned as a cloud or probability distribution function. The size and shape of this cloud is described by the wavefunction for the H- atom and it can be explicitly calculated from the Schrodinger equation: ψ ψ ψ 1 e + + ψ = m x y z 4πε 0 r Eψ (in 3D)
Electron Probability Distributions Examples of the 3-D probability distribution function ψ (electron cloud): The corresponding radial probability distribution function P(r): 4πε 0 a= = 5.9 10 me 11 m is the Bohr s radius which we have seen previously.
More Electron Probability Distributions
Quantum Number Recall that for a particle in a 1D box, we have one quantum number for the total energy of the particle. 5 λ / It arises from fitting the wavefunction [sin (nπx/l)] within a box of length L (quantization). λ 3 λ / λ λ / In the H-atom case, we are in 3D, the fitting of the wavefunction in space will result in additional quantum numbers (a total of 3).
Quantum Numbers 1. n Principle Quantum Number: related to the quantization of the main energy levels in the H-atom (as in the Bohr s model). E n 13.6eV =, n= 1,,3, n The other two related to the quantization of the orbital angular momentum of the electron. Only certain discrete values of the magnitude and the component of the orbital angular momentum are permitted:
Quantum Numbers. l Orbital Quantum Number: related to the quantization of the magnitude of the e - s orbital angular momentum L. ( 1 ), 0,1,,, ( 1) L= l l+ l = n (note: in Bohr s model, each energy level (n) corresponds to a single value of angular momentum. In the correct QM description, for each energy level (n), there are n possible values for L.) 3. m l Magnetic Quantum Number: related to the quantization of the direction of the e - s orbital angular momentum vector. L = m, m = 0, ± 1, ±,, ± l z l l (By convention, we pick the z-direction be the relevant direction for this quantization. Physically, there are no preference in the z-direction. The other two directions are not quantized.)
Magnetic Quantum Number Illustrations showing the relation between L and L z.
Electron Spins In 195, using again semi-classical model, Samuel Goudsmidt and George Uhlenbech demonstrate that this fine structure splitting is due to the spin angular momentum of the electron and this introduces the 4 th quantum number. 4. m s - Spin Quantum Number: The electron has another intrinsic physical characteristic akin to spin angular momentum associated with a spinning top. This quantum characteristic did not come out from Schrodinger s original theory. Its existence requires the consideration of relativistic quantum effects (Dirac s Theory). The direction of the spin angular momentum S z of the electron is quantized: 1 Sz = ms, ms = ± S= ss ( + 1), s= m s = 1 Pauli and Bohr
Wavefunction Labeling Scheme We have identified 4 separate quantum numbers for the H-atom (n, l, m l, m s ). For a given principal quantum number n, the H-atom has a given energy and there might be more than one distinct states (with additional choices for the other three quantum numbers). The fact that there are more than one distinct states for the same energy is call degeneracy. Historically, states with different principal quantum numbers are labeled as: n = 1: n = : n = 3: n = 4: K shell L shell M shell N shell And, states with different orbital quantum numbers are labeled as: l = 0: l = 1: l = : l = 3: l = 4: l = 5: s subshell p subshell d subshell f subshell g subshell h subshell
Wavefunction Labeling Scheme m l and m s are not labeled by this scheme.
Many Electron Atoms The Schrodinger equation for the general case with many electrons and protons interacting together quickly becomes very complicated. Central Field Approximation: Consider the effects from all electrons together as a spherically symmetric charge cloud so that each individual electron sees a total E field due to the nucleus + averaged-out cloud of all other electrons, In this approximation, U(r) is spherically symmetric (depends on r instead of all three spatial coordinates) This approximation is useful to understand the ground state of many electron atoms One can continue to use the 4 quantum numbers for the H-atom (n, l, m l, m s ) to describe them.
Pauli Exclusion Principle In order to understand the full electronic structures of the all elements beyond the simple single-electron H-atom, we need another quantum idea. In 195, Wolfgang Pauli proposed the Pauli s Exclusion Principle: no two electrons can occupy the same quantum-mechanical state in a given system, i.e., no two electrons in an atom can have the same set of value for all four quantum numbers (n, l, m l, m s ). The Pauli s Exclusion Principle + the set of the four quantum numbers give the complete prescription in identifying the ground state configuration of e - s for all elements in the Periodic Table. Then, all chemical properties for all atoms follow! Additional electrons cannot all crowded into the n = 1 state due to the Pauli s Exclusion Principle and they must distribute to other higher levels according to the ordering of the four quantum numbers.
Filling in the Ground State: Example H-atom (Z = 1 one e - ) E n = 1, l = 0 Helium (Z = two e - ) E n = 1, l = 0 1 filled, 1 free space the lowest level is now full Lithium (Z = 3 three e - ) n =, l = 1 n =, l = 0 n = 1, l = 0 m = 1 0 + 1 E l Last electron must go to n=, l=0 level by Pauli s Exclusion Principle. } n = level
Filling in the Ground State: Example Sodium (Z = 11) m = 1 0 + 1 l n = 3, l = 0 n =, l = 1 n =, l = 0 n = 1, l = 0 E
Spectroscopic Notation in the Periodic Table Typically, only the outer most shell (including the subshells within the outer most shell) is labeled. # of e - in that subshell H 1s shell n value 1 subshell l value He 1s 4 O 1s s p Z = 8 outer shell is n = two subshells (l =0 and l = 1) s p 8 electrons to fill, will fill K shell and 6 remaining will need to go to L shell: sl ( = 0) : ml = 0 pl ( = 1) : ml = 1, 0,1 only max slots 6 max slots with 4 taken
Ground-State Electron Configurations