PHYS 262. George Mason University. Professor Paul So
|
|
- Bennett Joseph
- 6 years ago
- Views:
Transcription
1 PHYS 6 George Mason University Professor Paul So
2 Chapter 40/41: Quantum Mechanics Wave Functions & 1D Schrodinger Eq Particle in a Box Wave function Energy levels Potential Wells/Barriers & Tunneling The Harmonic Oscillator The H-atom
3 Wave Equation for a Mechanical String y x, t For a wave on a string (1D) moving with speed v, a wave function must satisfy the wave equation (Ch. 15): y x, t, 1 y xt, y xt x v t It has the following sinusoidal form as its fundamental solution:, cos sin y xt A kx t B kx t where k is the wave number and f is the angular frequency of the wave. [A and B determines the amplitude and phase of the wave.]
4 Wave Equation for a String By substituting the fundamental wave function into the PDE, we can arrive at the algebraic relation (dispersion relation) that and k must satisfy: y x, t x k v or vk check Each spatial derivative of y xt, will pull out one k: Acos kx t Bsin kx t A sin kx t B cos x x k k k t So, the nd order spatial derivative gives, y x, t x k sin Ak kx t B kx t cos (Obviously, don t forget the signs.)
5 Wave Equation for a String check.... Each time derivative of y xt, will pull out one : y x, t t Acos kx t B sin kx t A sin kx t B cos kx t t So, the nd order time derivative gives, y x, t t sin A kx t B kx t cos Putting these back into the wave equation, we then have, (Again, don t forget the signs.), 1 yx, t y x t x v t k sin cos Ak kx t B kx t 1 cos v sin A kx t B kx t k v
6 Wave Equation for a String Putting the definitions for and k back into the dispersion relation, we have the familiar relation for wavelength, frequency, and wave speed. vk f v Thus, the fundamental property of a mechanical wave y xt, 1 y xt, is intimately linked to the form of the wave equation! x v t v f v f or k Now, we will try to use the same argument to find a wave equation for a quantum wave function. v
7 Wave Equation for a Quantum Free Particle A free particle has no force acting on it. Equivalently, the potential energy U( x) must be a constant for all x, i.e., F du x dx or U(x) is a constant. x 0 Since the reference point for U( x) is arbitrary, we can simply take U( x) 0. Then, the total energy E of a free particle will simply be its kinetic energy, E 1 mv p m m mv (non-relativistic) Now, from the de Broglie relations, the energy and momentum of this quantum free particle can be related to its wave number k and angular frequency through: h p E hf f E m k h h p k m
8 Wave Equation for a Quantum Free Particle Thus, a correct quantum wave function for this free particle must satisfy this quantum dispersion relation for k and : k * (non-relativistic) m We now assume the same fundamental sinusoidal form for the wave function of a quantum free particle with mass m, momentum p kand energy E : x, t Acoskx t B sinkx t Recall from our discussion on the mechanical wave, we have the following: x take out an overall k factor from x, t t take out an overall factor from x, t
9 Wave Equation for a Quantum Free Particle So, from the quantum dispersion relation, k m We can deduce that the PDE for the quantum wave function for this free particle must involves: x t Putting in other constants so that units are consistent and one additional dimensionless fitting constant C, we have, x, t x, t C m x t
10 Wave Equation for a Quantum Free Particle Now, we substitute our trial quantum wave function x, t Acoskx t B sinkx t into the proposed wave equation to solve for the fitting constant C: xt, cos m x m sin Ak kx t Bk kx t k Acos kx t B kx t m sin xt, C CA kx t B kx t t sin cos sin CB cos kx t CA kx t
11 Wave Equation for a Quantum Free Particle k Equating the two terms and using the equality, we have, m k m sin Acos kx t B kx t CB cos kx t CAsin kx t A CB B CA Substituting the first eq into the second, we have, In order for this equality to be true for all xt,, all coeff s for cos and sin must equal to each other, BC CB C 1 Thus, the fitting constant is C i where i 1.
12 Wave Equation for a Quantum Free Particle Then, finally, putting everything together, we have the desired wave equation for a quantum free particle, x, t x, t i m x t This is the 1D Schrodinger s Equation for a free particle. With B CAiA, the free particle quantum wave function can also be written in a compact exponential form using the Euler s formula, x, t Acoskx t isinkx t xt, Ae ikxt (quantum wave function for a free particle)
13 Free Particle Wave Function & Uncertainty Principle The wave function for a free particle is a complex function with sinusoidal real and imaginary parts A quantum free particle exists in all space,, x & t (wave function extends into all space & time) but p0 & E 0 (energy and momentum is fixed) Note: x p & t E can still be satisfied.
14 More Realistic Particle (Wave Packets) Under more practical circumstance, a particle will have a relatively well defined position and momentum so that both x and p will be finite with limited spatial extents. A more localized quantum particle can not be a pure sine wave and it must be described by a wave packet with a combination of many sine waves. (, ) ( ) ikx xt Ake t dk (a linear combination of many sine waves.) The coefficient A(k) gives the relative proportion of the various sine waves with diff. k (wave number).
15 Wave Packets Recall: Combination of two sine waves more localized than a pure sine wave.
16 Wave Packets (characteristic) p smaller x bigger
17 Wave Packets (characteristic) p bigger x smaller The is consistent with: x p!
18 Quantum Wave Function In QM, the matter wave postulated by de Broglie is described by a complexvalued wavefunction (x,t) which is the fundamental descriptor for a quantum particle. 1. Its absolute value squared ( xt, ) dx gives the probability of finding the particle in an infinitesimal volume dx at time t.. For any Q problem: The goal is to find ( xt, ) for the particle for all time. Physical interactions involves operations (O) on this wave function: O( x, t) Experimental measurements will involve the products, ( xt, ) O( x, t) Re/Im (x,t) x,t (x,t) is a complex-valued function of space and time.
19 The 1D Schrodinger Equation As we have see, ( x, t) ( x, t) U( x) ( x, t) i m x t KE + PE = Total E - the first term ( nd order spatial derivative term) in the Schrodinger equation is associated with the Kinetic Energy of the particle - the last term (the 1 st order time derivative term) is associated with the total energy of the particle - together with the Potential Energy term U(x)(x) the Schrodinger equation is basically a statement on the conservation of energy.
20 The Schrodinger Equation In Classical Mechanics, we have the Newton s equation which describes the trajectory x(t) of a particle: F mx In EM, we have the wave equation for the propagation of the E, B fields: EB, 1 EB, x c t (derived from Maxwell s eqns) In QM, Schrodinger equation prescribes the evolution of the wavefunction for a particle in time t and space x under the influence of a potential energy U(x), U(x) ( x, t) ( x, t) U( x) ( x, t) i m x t (general 1D Schrödinger equation)
21 Wave Function and Probability * ( xt, ) ( xt, ) ( xt, ) is the probability distribution function for the quantum particle. In other words, ( x, t) (shaded area) is the probability in finding the particle in the interval [ xx, dx] at time t. dx Since p( xdx ) ( xt, ) dxis a probability, it has to be normalized! p( x) dx ( x, t) dx 1 (At any instance of time t, the particle must be somewhere in space!)
22 Stationary States For most problems, we can factor out the time dependence by assuming the following harmonic form for the time dependence, i t ( xt, ) ( xe ) ikxt ikx it (Recall the free particle case: ( x, t) Ae Ae e.) With E /, we can rewrite the time exponent in terms of E, ( xt, ) ( xe ) iet / ( xt, ) is a state with a definite energy E and is called a stationary state. ( x) is called the time-independent wave function.
23 The Time-Independent Schrödinger Equation Substituting this factorization into the general time-dependent Schrodinger Eq, we have RHS and, LHS ( xt, ) ie i i ( x) e i ( x) e E ( x) e t t ( xt, ) d ( x) iet/ e x dx iet / iet / iet / d ( x) iet/ e iet / iet/ U( x ) ( x) e E ( x) e m dx d ( x) U( x) ( x) E( x) m dx (time dependence can be cancelled out!) (time-independent Schrodinger equation)
24 More on (time-independent) Wavefunction Note that, ( xt, ) * * iet/ iet/ ( xt, ) ( xt, ) ( xe ) ( x) e ( x) ( x) e ( x) ( x) ( x) * i( Et/ Et/ ) * So, in general, the probability in finding the particle in the interval [a,b] is given by: b pab ( x) dx a p(x) Note: ( x) is not the probability density ( x) is the probability density. a b x
25 More on (time-independent) Wavefunction Other physical observables can be obtained from (x) by the following operation: example (position x): x xp( x) dx x ( x) dx - x is called the expectation value (of x): it is the experimental value that one should expect to measure in real experiments! In general, any experimental observable (position, momentum, energy, etc.) O(x) will have an expectation value given by: O O( x) ( x) dx O can be x, p, E, etc. Note: Expectation values of physically measurable functions are the only experimentally accessible quantities in QM. Wavefunction ( x) itself is not a physically measureable quantity.
26 Solving QM Problems with (timeindependent) Schrodinger Equation Given: A particle is moving under the influence of a potential U(x). Examples: Free particle: U(x) = 0 Particle in a box: U( x) 0, 0 x L, elsewhere Barrier: U( x) U, 0 0 x L 0, elsewhere HMO: 1 U( x) k' x
27 Solving QM Problems with (timeindependent) Schrodinger Equation Solve time-independent Schrodinger equation for (x) as a function of energy E, with the restrictions: d ( x) ( x) and are continuous everywhere for smooth U(x). dx ( x) is normalized, i.e., ( x) dx 1 Bounded solution: ( x) 0 as x Then, expectation values of physical measurable quantities can be calculated.
28 Particle in a Box Classical Picture A 1-D box with hard walls: U(0) U( L) (non-penetrable) A free particle inside the box: U( x) 0 (inside box) No forces acting on the particle except at hard walls. P (in x) is conserved between bounces P is fixed but P switches sign between bounces.
29 Particle in a Box (Quantum Picture) The situation can be described by the following potential energy U(x): U( x) 0, 0 x L, elsewhere The time-independent Schrodinger equation is: d ( x) U( x) ( x) E( x) m dx Recall, this is basically KE + PE = Total E Problem statement: For this U(x), what are the possible wave functions (x) and their corresponding allowed energies E?
30 Wave functions for a Particle in a Box Inside the box, 0 x L, U(x) = 0, and the particle is free. From before, we know that the wave function for a free particle has the following form: ikx ikx (linear combination of the two inside ( x) Ae 1 Ae possible solutions.) p k where A 1 and A are constants that will be determined later. E m m Outside the box, U( x), and the particle cannot exist outside the box and ( x) 0 (outside the box) outside At the boundary, x = 0 and x = L,the wavefunction has to be continuous: inside (0, L) (0, L) 0 outside
31 Wavefunctions for a Particle in a Box Let see how this boundary condition imposes restrictions on the two constants, A 1 and A, for the wave function. Using the Euler s formula, we can rewrite the interior wave function in terms of sine and cosine: ikx e 1 A A cos kx ia A sin kx inside( x) A cos kx i sin kx A cos kx i sin kx 1 1 Imposing the boundary condition at x = 0, (0) cos0 sin 0 A A inside A A i A A 1 1 ikx e 1 0 A A 1 (where C=iA 1 ) inside( x) ia1 sinkx Csinkx
32 Wavefunctions for a Particle in a Box Now, consider the boundary condition at x = L: ( L inside ) C sin kl 0 For a non-trivial solution ( C0), only certain sine waves with a particular choice of wave numbers (k) can satisfy this condition: n kl n n or kn, n1,,3, L This implies that the wavelengths within the box is quantized! n k n L L, n 1,,3, n n Allowed wavefunctions must have wavenlengthes exactly fit within the box!
33 Wavefunction for a Particle in a Box Rewriting this, we have, L n, n 1,,3, n Graphically, it looks like Since k n is quantized, only a discrete set of ( x n ) is allowed as solutions, n n( x) CsinknxCsin x, L n 1,,3, 5 / 3 / / (similar to standing waves on a cramped string)
34 Quantized Energies for a Particle in a Box Since the wave number k n is quantized, the energy for the particle in the box is also quantized: E n kn n n n h or, n1,,3, m m L ml 8mL (n is called the quantum number) Note: the lowest energy is not zero: h E1 0 8mL n = 0 gives (x) = 0 and it means no particle.
35 Probability and Wavefunction Recall that (x) (and not the wavefunction itself (x)) is the probability density function. In particular, ( ) sin n x x dx C dx L gives the probability in finding the particle in an interval [x, x+dx] within the box.
36 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. 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.
37 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, ( x) dx 1 (normalization condition) L n L C sin xdxc 1 L 0 So, the normalization condition fixes the final free constant C in the wavefunction, C L. This then gives, ( ) sin n x n x (particle in a box) L L
38 Time Dependence Note that with (x) found, we can write down the full wavefunction for the time-dependent Schrodinger equation as: ( xt, ) ( xe ) iet n n / recall E hf n x n( xt, ) sin e L L iet / / Note that the absolute value for e iet is unity, i.e., iet / iet / iet / 0 e e e e 1 so that n (x,t) = (x) is independent of time and probability density in finding the particle in the box is also independent of time.
39 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 elsewhere 0, 0 x L 0, For a finite square-well, there will typically be only a finite number of bound states.
40 Finite Square-Well Potential Similar to the particle in a box problem, U(x) = 0 inside the well, we have or, d ( x) me k ( x), where k dx ( x) Ae Ae ikx 1 ikx inside( x) AcoskxBsinkx 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 ( x) dx m mu 0 E U 0 E ( x) ( x), where (outside the well)
41 Finite Square-Well Potential Since U 0 > E, is real and the wavefunction outside the well is given in terms of exponentials instead of harmonic functions: x ( x) Ce De x 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 x ( ) x and ( ) 0 x Ce xl x De x
42 Finite Square-Well Potential For a physical quantum particle, both ( x) and d ( x)/ dx must be continuous at x = 0 and x = L. Matching x0 ( x), inside( x),and xl( 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.
43 Example: e in a Square-Well/Quantum Dot An electron trapped in a Square-Well potential with width L = 0.5nm (~size of an atom) a) What is ground state energy E if this well is infinitely deep U 0 = instead? E, Js 31 9 ml kg m,1 J ev b) Now, back to a finite well with U 6E 9.0eV 0,1 The energy levels for the finite well are given as shown on the next slide. (not derived here)
44 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 hc 140eV s 460nm E E 3.6eV 0.94eV 1
45 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
46 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!
47 Tunneling Through a Barrier Quantum Picture: x < 0 and x > L (free space): 0 x L(inside the barrier): The wavefunction for a free particle with definite E and P is sinusoidal, e ikx or e -ikx. E U 0 wavefunction is a decaying exponential e -x. exponential function within barrier
48 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 L E E T Ge,, G 16 1 U0 U0 (for E/U 0 small)
49 Quantum Tunneling A. I. Kolesnikov et al., Phys. Rev. Lett. (016) A ring-shaped new molecular state for water when a single water molecule is confined inside a hexagonally shaped channel of the gemstone beryl. The H atom delocalizes (spread out) by tunneling to other five classically inaccessible orientational states.
50 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.
51 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. k' m
52 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.
53 The Harmonic Oscillator For the Quantum analysis, we will use the same form of the potential energy for a quantum Harmonic Oscillator. d m dx ( x) 1 k ' x ( x ) E ( x ) And we have the following quantized energies: d ( x) m1 k ' x E ( x ) The solutions for this ordinary differential equation with the boundary condition ( x) 0 as x are called the Hermite functions: or 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. ( x) Ce mk ' x dx 1 En n, n0,1,, (ground state n=0)
54 The Harmonic Oscillator Hermite Functions 1 En n, n0,1,, note: wavefunction penetration into classically forbidden regions. note: similar to previous examples, the lowest E state is not zero.
55 The Harmonic Oscillator Probability Distribution Function: Classically, the particle with energy E will slow down as it climbs up on both side of the potential hills and it will spend most of its time near. x A The blue curve depicts this classical behavior and the QM ~ CM as the quantum number n increases.
56 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 (), 4 r 0 r x y z is the radius in spherical coordinates.
57 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: 1 e m x y z 4 0 r E (in 3D)
58 Electron Probability Distributions In 3D, the probability in finding the electron in a given volume dv is given by, ( xyz,, ) dv A good way to visualize this 3D probability distribution is to consider a thin spherical shell with radius r and thickness dr as our choice for dv: dv 4 r dr dr r We denote the probability of finding the electron within this thin radial shell as the radial probability distribution function P(r) with: P() rdr dv 4rdr
59 Electron Probability Distributions Examples of the 3-D probability distribution function (electron cloud): The corresponding radial probability distribution function P(r): 4 0 a me 11 m is the Bohr s radius which we have seen previously.
60 More Electron Probability Distributions
61 Quantum Number Recall that for a particle in a 1D box, we have one quantum number for the total energy of the particle. 5 / It arises from fitting the wavefunction [sin (nx/l)] within a box of length L (quantization). 3 / / 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).
62 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, n1,,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:
63 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.)
64 Magnetic Quantum Number Illustrations showing the relation between L and L z.
65 Zeeman Effect Experimentally, it was found that under a magnetic field, the energy levels of the H-atom will split according to the magnetic quantum number m l. Semi-classical explanation: L B e - e - orbits around the nucleus and it forms a current loop. L z measures the orientation of L with respect to B and thus affects the energy level of the H-atom.
66 Anomalous Zeeman Effect Predicted with alone m l Additional experiments shows that some of the Zeeman lines are further split.
67 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. 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: S m, m z s s 1 S s( s1), s m s 1 Pauli and Bohr
68 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: n1: 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
69 Wavefunction Labeling Scheme m l and m s are not labeled by this scheme.
70 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.
71 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.
72 Filling in the Ground State: Example H-atom (Z = 1 one e - ) n = 1, l = 0 E Helium (Z = two e - ) n = 1, l = 0 E 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 E m l Last electron must go to n=, l=0 level by Pauli s Exclusion Principle. n = level
73 Filling in the Ground State: Example Sodium (Z = 11) m l n = 3, l = 0 n =, l = 1 n =, l = 0 n = 1, l = 0 E
74 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
75 Ground-State Electron Configurations
Probability and Normalization
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
More information* = 2 = Probability distribution function. probability of finding a particle near a given point x,y,z at a time t
Quantum Mechanics Wave functions and the Schrodinger equation Particles behave like waves, so they can be described with a wave function (x,y,z,t) A stationary state has a definite energy, and can be written
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
More informationQuantum Mechanics. p " The Uncertainty Principle places fundamental limits on our measurements :
Student Selected Module 2005/2006 (SSM-0032) 17 th November 2005 Quantum Mechanics Outline : Review of Previous Lecture. Single Particle Wavefunctions. Time-Independent Schrödinger equation. Particle in
More informationModern Physics for Scientists and Engineers International Edition, 4th Edition
Modern Physics for Scientists and Engineers International Edition, 4th Edition http://optics.hanyang.ac.kr/~shsong Review: 1. THE BIRTH OF MODERN PHYSICS 2. SPECIAL THEORY OF RELATIVITY 3. THE EXPERIMENTAL
More informationIntroduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world,
Introduction to Quantum Mechanics (Prelude to Nuclear Shell Model) Heisenberg Uncertainty Principle In the microscopic world, x p h π If you try to specify/measure the exact position of a particle you
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 6.2 6.3 6.4 6.5 6.6 6.7 The Schrödinger Wave Equation Expectation Values Infinite Square-Well Potential Finite Square-Well Potential Three-Dimensional Infinite-Potential
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
More informationWave nature of particles
Wave nature of particles We have thus far developed a model of atomic structure based on the particle nature of matter: Atoms have a dense nucleus of positive charge with electrons orbiting the nucleus
More informationA more comprehensive theory was needed. 1925, Schrödinger and Heisenberg separately worked out a new theory Quantum Mechanics.
Ch28 Quantum Mechanics of Atoms Bohr s model was very successful to explain line spectra and the ionization energy for hydrogen. However, it also had many limitations: It was not able to predict the line
More informationA few principles of classical and quantum mechanics
A few principles of classical and quantum mechanics The classical approach: In classical mechanics, we usually (but not exclusively) solve Newton s nd law of motion relating the acceleration a of the system
More informationSemiconductor Physics and Devices
Introduction to Quantum Mechanics In order to understand the current-voltage characteristics, we need some knowledge of electron behavior in semiconductor when the electron is subjected to various potential
More informationCHAPTER 8 The Quantum Theory of Motion
I. Translational motion. CHAPTER 8 The Quantum Theory of Motion A. Single particle in free space, 1-D. 1. Schrodinger eqn H ψ = Eψ! 2 2m d 2 dx 2 ψ = Eψ ; no boundary conditions 2. General solution: ψ
More informationCHAPTER 6 Quantum Mechanics II
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite-Potential Well
More informationFinal Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall Duration: 2h 30m
Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. ------------------- Duration: 2h 30m Chapter 39 Quantum Mechanics of Atoms Units of Chapter 39 39-1 Quantum-Mechanical View of Atoms 39-2
More informationThe Hydrogen Atom. Dr. Sabry El-Taher 1. e 4. U U r
The Hydrogen Atom Atom is a 3D object, and the electron motion is three-dimensional. We ll start with the simplest case - The hydrogen atom. An electron and a proton (nucleus) are bound by the central-symmetric
More informationPHYS 3313 Section 001 Lecture # 22
PHYS 3313 Section 001 Lecture # 22 Dr. Barry Spurlock Simple Harmonic Oscillator Barriers and Tunneling Alpha Particle Decay Schrodinger Equation on Hydrogen Atom Solutions for Schrodinger Equation for
More informationLecture 13: Barrier Penetration and Tunneling
Lecture 13: Barrier Penetration and Tunneling nucleus x U(x) U(x) U 0 E A B C B A 0 L x 0 x Lecture 13, p 1 Today Tunneling of quantum particles Scanning Tunneling Microscope (STM) Nuclear Decay Solar
More informationFinal Exam. Tuesday, May 8, Starting at 8:30 a.m., Hoyt Hall.
Final Exam Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. Summary of Chapter 38 In Quantum Mechanics particles are represented by wave functions Ψ. The absolute square of the wave function Ψ 2
More informationElectron in a Box. A wave packet in a square well (an electron in a box) changing with time.
Electron in a Box A wave packet in a square well (an electron in a box) changing with time. Last Time: Light Wave model: Interference pattern is in terms of wave intensity Photon model: Interference in
More informationQuantum Mechanics. The Schrödinger equation. Erwin Schrödinger
Quantum Mechanics The Schrödinger equation Erwin Schrödinger The Nobel Prize in Physics 1933 "for the discovery of new productive forms of atomic theory" The Schrödinger Equation in One Dimension Time-Independent
More informationComplete nomenclature for electron orbitals
Complete nomenclature for electron orbitals Bohr s model worked but it lacked a satisfactory reason why. De Broglie suggested that all particles have a wave nature. u l=h/p Enter de Broglie again It was
More informationOpinions on quantum mechanics. CHAPTER 6 Quantum Mechanics II. 6.1: The Schrödinger Wave Equation. Normalization and Probability
CHAPTER 6 Quantum Mechanics II 6.1 The Schrödinger Wave Equation 6. Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite- 6.6 Simple Harmonic
More informationPHYS 3313 Section 001 Lecture #20
PHYS 3313 Section 001 ecture #0 Monday, April 10, 017 Dr. Amir Farbin Infinite Square-well Potential Finite Square Well Potential Penetration Depth Degeneracy Simple Harmonic Oscillator 1 Announcements
More information8 Wavefunctions - Schrödinger s Equation
8 Wavefunctions - Schrödinger s Equation So far we have considered only free particles - i.e. particles whose energy consists entirely of its kinetic energy. In general, however, a particle moves under
More informationComplementi di Fisica Lectures 10-11
Complementi di Fisica - Lectures 1-11 15/16-1-1 Complementi di Fisica Lectures 1-11 Livio Lanceri Università di Trieste Trieste, 15/16-1-1 Course Outline - Reminder Quantum Mechanics: an introduction Reminder
More informationThere is light at the end of the tunnel. -- proverb. The light at the end of the tunnel is just the light of an oncoming train. --R.
A vast time bubble has been projected into the future to the precise moment of the end of the universe. This is, of course, impossible. --D. Adams, The Hitchhiker s Guide to the Galaxy There is light at
More informationAtoms. Radiation from atoms and molecules enables the most accurate time and length measurements: Atomic clocks
Atoms Quantum physics explains the energy levels of atoms with enormous accuracy. This is possible, since these levels have long lifetime (uncertainty relation for E, t). Radiation from atoms and molecules
More informationSolving the Schrodinger Equation
Time-dependent Schrödinger Equation: i!!!2 " (x,t) =!t 2m! 2 " (x,t) + U(x)" (x,t) 2!x Stationary Solutions:! (x,t) = "(x)(t)!(t) = e "it, = E! Time-independent Schrödinger equation:!!2 2m d 2 "(x) + U(x)"(x)
More informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)
More informationDept. of Physics, MIT Manipal 1
Chapter 1: Optics 1. In the phenomenon of interference, there is A Annihilation of light energy B Addition of energy C Redistribution energy D Creation of energy 2. Interference fringes are obtained using
More informationCHAPTER 2: POSTULATES OF QUANTUM MECHANICS
CHAPTER 2: POSTULATES OF QUANTUM MECHANICS Basics of Quantum Mechanics - Why Quantum Physics? - Classical mechanics (Newton's mechanics) and Maxwell's equations (electromagnetics theory) can explain MACROSCOPIC
More informationLecture 10: The Schrödinger Equation Lecture 10, p 1
Lecture 10: The Schrödinger Equation Lecture 10, p 1 Overview Probability distributions Schrödinger s Equation Particle in a Bo Matter waves in an infinite square well Quantized energy levels y() U= n=1
More informationQuantum Mechanics & Atomic Structure (Chapter 11)
Quantum Mechanics & Atomic Structure (Chapter 11) Quantum mechanics: Microscopic theory of light & matter at molecular scale and smaller. Atoms and radiation (light) have both wave-like and particlelike
More informationChapter 4 (Lecture 6-7) Schrodinger equation for some simple systems Table: List of various one dimensional potentials System Physical correspondence
V, E, Chapter (Lecture 6-7) Schrodinger equation for some simple systems Table: List of various one dimensional potentials System Physical correspondence Potential Total Energies and Probability density
More informationatoms and light. Chapter Goal: To understand the structure and properties of atoms.
Quantum mechanics provides us with an understanding of atomic structure and atomic properties. Lasers are one of the most important applications of the quantummechanical properties of atoms and light.
More informationQuantum Mechanics of Atoms
Quantum Mechanics of Atoms Your theory is crazy, but it's not crazy enough to be true N. Bohr to W. Pauli Quantum Mechanics of Atoms 2 Limitations of the Bohr Model The model was a great break-through,
More informationComplementi di Fisica Lectures 5, 6
Complementi di Fisica - Lectures 5, 6 9/3-9-15 Complementi di Fisica Lectures 5, 6 Livio Lanceri Università di Trieste Trieste, 9/3-9-15 Course Outline - Reminder Quantum Mechanics: an introduction Reminder
More informationCOLLEGE PHYSICS. Chapter 30 ATOMIC PHYSICS
COLLEGE PHYSICS Chapter 30 ATOMIC PHYSICS Matter Waves: The de Broglie Hypothesis The momentum of a photon is given by: The de Broglie hypothesis is that particles also have wavelengths, given by: Matter
More informationAe ikx Be ikx. Quantum theory: techniques and applications
Quantum theory: techniques and applications There exist three basic modes of motion: translation, vibration, and rotation. All three play an important role in chemistry because they are ways in which molecules
More informationCHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 5.1 X-Ray Scattering 5. De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles? 5.6 Uncertainty Principle Many experimental
More informationThe Schrödinger Equation in One Dimension
The Schrödinger Equation in One Dimension Introduction We have defined a comple wave function Ψ(, t) for a particle and interpreted it such that Ψ ( r, t d gives the probability that the particle is at
More informationChapter 7. Bound Systems are perhaps the most interesting cases for us to consider. We see much of the interesting features of quantum mechanics.
Chapter 7 In chapter 6 we learned about a set of rules for quantum mechanics. Now we want to apply them to various cases and see what they predict for the behavior of quanta under different conditions.
More informationECE440 Nanoelectronics. Lecture 07 Atomic Orbitals
ECE44 Nanoelectronics Lecture 7 Atomic Orbitals Atoms and atomic orbitals It is instructive to compare the simple model of a spherically symmetrical potential for r R V ( r) for r R and the simplest hydrogen
More informationCHAPTER 28 Quantum Mechanics of Atoms Units
CHAPTER 28 Quantum Mechanics of Atoms Units Quantum Mechanics A New Theory The Wave Function and Its Interpretation; the Double-Slit Experiment The Heisenberg Uncertainty Principle Philosophic Implications;
More informationLecture 10: The Schrödinger Equation. Lecture 10, p 2
Quantum mechanics is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that
More informationOld and new quantum theory
Old and new quantum theory Faults of the Bohr model: - gives only position of the lines and not the intensity - does not explain the number of electrons on each orbit - gives innacurate results for atoms
More informationChapter 6. Quantum Theory of the Hydrogen Atom
Chapter 6 Quantum Theory of the Hydrogen Atom 1 6.1 Schrodinger s Equation for the Hydrogen Atom Symmetry suggests spherical polar coordinates Fig. 6.1 (a) Spherical polar coordinates. (b) A line of constant
More informationBohr s Correspondence Principle
Bohr s Correspondence Principle In limit that n, quantum mechanics must agree with classical physics E photon = 13.6 ev 1 n f n 1 i = hf photon In this limit, n i n f, and then f photon electron s frequency
More informationDavid J. Starling Penn State Hazleton PHYS 214
Not all chemicals are bad. Without chemicals such as hydrogen and oxygen, for example, there would be no way to make water, a vital ingredient in beer. -Dave Barry David J. Starling Penn State Hazleton
More informationLecture 10: The Schrödinger Equation. Lecture 10, p 2
Quantum mechanics is the description of the behavior of matter and light in all its details and, in particular, of the happenings on an atomic scale. Things on a very small scale behave like nothing that
More informationPhysics 43 Exam 2 Spring 2018
Physics 43 Exam 2 Spring 2018 Print Name: Conceptual Circle the best answer. (2 points each) 1. Quantum physics agrees with the classical physics limit when a. the total angular momentum is a small multiple
More informationUnbound States. 6.3 Quantum Tunneling Examples Alpha Decay The Tunnel Diode SQUIDS Field Emission The Scanning Tunneling Microscope
Unbound States 6.3 Quantum Tunneling Examples Alpha Decay The Tunnel Diode SQUIDS Field Emission The Scanning Tunneling Microscope 6.4 Particle-Wave Propagation Phase and Group Velocities Particle-like
More informationSparks CH301. Quantum Mechanics. Waves? Particles? What and where are the electrons!? UNIT 2 Day 3. LM 14, 15 & 16 + HW due Friday, 8:45 am
Sparks CH301 Quantum Mechanics Waves? Particles? What and where are the electrons!? UNIT 2 Day 3 LM 14, 15 & 16 + HW due Friday, 8:45 am What are we going to learn today? The Simplest Atom - Hydrogen Relate
More informationApplications of Quantum Theory to Some Simple Systems
Applications of Quantum Theory to Some Simple Systems Arbitrariness in the value of total energy. We will use classical mechanics, and for simplicity of the discussion, consider a particle of mass m moving
More informationChapter. 5 Bound States: Simple Case
Announcement Course webpage http://highenergy.phys.ttu.edu/~slee/2402/ Textbook PHYS-2402 Lecture 12 HW3 (due 3/2) 13, 15, 20, 31, 36, 41, 48, 53, 63, 66 ***** Exam: 3/12 Ch.2, 3, 4, 5 Feb. 26, 2015 Physics
More informationPHY 114 A General Physics II 11 AM-12:15 PM TR Olin 101
PHY 114 A General Physics II 11 AM-1:15 PM TR Olin 101 Plan for Lecture 3 (Chapter 40-4): Some topics in Quantum Theory 1. Particle behaviors of electromagnetic waves. Wave behaviors of particles 3. Quantized
More informationLine spectrum (contd.) Bohr s Planetary Atom
Line spectrum (contd.) Hydrogen shows lines in the visible region of the spectrum (red, blue-green, blue and violet). The wavelengths of these lines can be calculated by an equation proposed by J. J. Balmer:
More informationModern physics. 4. Barriers and wells. Lectures in Physics, summer
Modern physics 4. Barriers and wells Lectures in Physics, summer 016 1 Outline 4.1. Particle motion in the presence of a potential barrier 4.. Wave functions in the presence of a potential barrier 4.3.
More informationQuantum Theory. Thornton and Rex, Ch. 6
Quantum Theory Thornton and Rex, Ch. 6 Matter can behave like waves. 1) What is the wave equation? 2) How do we interpret the wave function y(x,t)? Light Waves Plane wave: y(x,t) = A cos(kx-wt) wave (w,k)
More informationA 2 sin 2 (n x/l) dx = 1 A 2 (L/2) = 1
VI 15 Model Problems 014 - Particle in box - all texts, plus Tunneling, barriers, free particle Atkins(p.89-300),ouse h.3 onsider E-M wave first (complex function, learn e ix form) E 0 e i(kx t) = E 0
More informationPhysics 486 Discussion 5 Piecewise Potentials
Physics 486 Discussion 5 Piecewise Potentials Problem 1 : Infinite Potential Well Checkpoints 1 Consider the infinite well potential V(x) = 0 for 0 < x < 1 elsewhere. (a) First, think classically. Potential
More informationFinal Exam: Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall.
Final Exam: Tuesday, May 8, 2012 Starting at 8:30 a.m., Hoyt Hall. Chapter 38 Quantum Mechanics Units of Chapter 38 38-1 Quantum Mechanics A New Theory 37-2 The Wave Function and Its Interpretation; the
More informationC/CS/Phys C191 Particle-in-a-box, Spin 10/02/08 Fall 2008 Lecture 11
C/CS/Phys C191 Particle-in-a-box, Spin 10/0/08 Fall 008 Lecture 11 Last time we saw that the time dependent Schr. eqn. can be decomposed into two equations, one in time (t) and one in space (x): space
More informationLecture 12: Particle in 1D boxes & Simple Harmonic Oscillator
Lecture 12: Particle in 1D boxes & Simple Harmonic Oscillator U(x) E Dx y(x) x Dx Lecture 12, p 1 Properties of Bound States Several trends exhibited by the particle-in-box states are generic to bound
More informationPH 253 Final Exam: Solution
PH 53 Final Exam: Solution 1. A particle of mass m is confined to a one-dimensional box of width L, that is, the potential energy of the particle is infinite everywhere except in the interval 0
More information2.4. Quantum Mechanical description of hydrogen atom
2.4. Quantum Mechanical description of hydrogen atom Atomic units Quantity Atomic unit SI Conversion Ang. mom. h [J s] h = 1, 05459 10 34 Js Mass m e [kg] m e = 9, 1094 10 31 kg Charge e [C] e = 1, 6022
More informationPhysics-I. Dr. Anurag Srivastava. Web address: Visit me: Room-110, Block-E, IIITM Campus
Physics-I Dr. Anurag Srivastava Web address: http://tiiciiitm.com/profanurag Email: profanurag@gmail.com Visit me: Room-110, Block-E, IIITM Campus Syllabus Electrodynamics: Maxwell s equations: differential
More informationChapter 7 QUANTUM THEORY & ATOMIC STRUCTURE Brooks/Cole - Thomson
Chapter 7 QUANTUM THEORY & ATOMIC STRUCTURE 1 7.1 The Nature of Light 2 Most subatomic particles behave as PARTICLES and obey the physics of waves. Light is a type of electromagnetic radiation Light consists
More informationFinal Exam - Solutions PHYS/ECE Fall 2011
Final Exam - Solutions PHYS/ECE 34 - Fall 211 Problem 1 Cosmic Rays The telescope array project in Millard County, UT can detect cosmic rays with energies up to E 1 2 ev. The cosmic rays are of unknown
More informationThe Wave Function. Chapter The Harmonic Wave Function
Chapter 3 The Wave Function On the basis of the assumption that the de Broglie relations give the frequency and wavelength of some kind of wave to be associated with a particle, plus the assumption that
More informationThe Wave Function. Chapter The Harmonic Wave Function
Chapter 3 The Wave Function On the basis of the assumption that the de Broglie relations give the frequency and wavelength of some kind of wave to be associated with a particle, plus the assumption that
More informationChem 467 Supplement to Lecture 19 Hydrogen Atom, Atomic Orbitals
Chem 467 Supplement to Lecture 19 Hydrogen Atom, Atomic Orbitals Pre-Quantum Atomic Structure The existence of atoms and molecules had long been theorized, but never rigorously proven until the late 19
More informationNotes for Special Relativity, Quantum Mechanics, and Nuclear Physics
Notes for Special Relativity, Quantum Mechanics, and Nuclear Physics 1. More on special relativity Normally, when two objects are moving with velocity v and u with respect to the stationary observer, the
More informationQuantum Mechanical Tunneling
The square barrier: Quantum Mechanical Tunneling Behaviour of a classical ball rolling towards a hill (potential barrier): If the ball has energy E less than the potential energy barrier (U=mgy), then
More informationChapter 28 Quantum Theory Lecture 24
Chapter 28 Quantum Theory Lecture 24 28.1 Particles, Waves, and Particles-Waves 28.2 Photons 28.3 Wavelike Properties Classical Particles 28.4 Electron Spin 28.5 Meaning of the Wave Function 28.6 Tunneling
More informationPhysics 2203, 2011: Equation sheet for second midterm. General properties of Schrödinger s Equation: Quantum Mechanics. Ψ + UΨ = i t.
General properties of Schrödinger s Equation: Quantum Mechanics Schrödinger Equation (time dependent) m Standing wave Ψ(x,t) = Ψ(x)e iωt Schrödinger Equation (time independent) Ψ x m Ψ x Ψ + UΨ = i t +UΨ
More informationAtomic Structure and Processes
Chapter 5 Atomic Structure and Processes 5.1 Elementary atomic structure Bohr Orbits correspond to principal quantum number n. Hydrogen atom energy levels where the Rydberg energy is R y = m e ( e E n
More informationMathematical Tripos Part IB Michaelmas Term Example Sheet 1. Values of some physical constants are given on the supplementary sheet
Mathematical Tripos Part IB Michaelmas Term 2015 Quantum Mechanics Dr. J.M. Evans Example Sheet 1 Values of some physical constants are given on the supplementary sheet 1. Whenasampleofpotassiumisilluminatedwithlightofwavelength3
More information2. The Schrödinger equation for one-particle problems. 5. Atoms and the periodic table of chemical elements
1 Historical introduction The Schrödinger equation for one-particle problems 3 Mathematical tools for quantum chemistry 4 The postulates of quantum mechanics 5 Atoms and the periodic table of chemical
More informationLecture 5. Potentials
Lecture 5 Potentials 51 52 LECTURE 5. POTENTIALS 5.1 Potentials In this lecture we will solve Schrödinger s equation for some simple one-dimensional potentials, and discuss the physical interpretation
More informationQuantum Mechanics. Watkins, Phys 365,
Quantum Mechanics Objectives: quantitative description of the behavior of nature at the atomic scale Central Idea: Wave-particle duality Particles obeyed classical physics: discrete, indivisible, could
More informationChapter 8 Chapter 8 Quantum Theory: Techniques and Applications (Part II)
Chapter 8 Chapter 8 Quantum Theory: Techniques and Applications (Part II) The Particle in the Box and the Real World, Phys. Chem. nd Ed. T. Engel, P. Reid (Ch.16) Objectives Importance of the concept for
More informationProblem Set 5: Solutions
University of Alabama Department of Physics and Astronomy PH 53 / eclair Spring 1 Problem Set 5: Solutions 1. Solve one of the exam problems that you did not choose.. The Thompson model of the atom. Show
More information4/21/2010. Schrödinger Equation For Hydrogen Atom. Spherical Coordinates CHAPTER 8
CHAPTER 8 Hydrogen Atom 8.1 Spherical Coordinates 8.2 Schrödinger's Equation in Spherical Coordinate 8.3 Separation of Variables 8.4 Three Quantum Numbers 8.5 Hydrogen Atom Wave Function 8.6 Electron Spin
More informationPhysics 102: Lecture 24. Bohr vs. Correct Model of Atom. Physics 102: Lecture 24, Slide 1
Physics 102: Lecture 24 Bohr vs. Correct Model of Atom Physics 102: Lecture 24, Slide 1 Plum Pudding Early Model for Atom positive and negative charges uniformly distributed throughout the atom like plums
More informationU(x) Finite Well. E Re ψ(x) Classically forbidden
Final Exam Physics 2130 Modern Physics Tuesday December 18, 2001 Point distribution: All questions are worth points 8 points. Answers should be bubbled onto the answer sheet. 1. At what common energy E
More informationChapter 28 Quantum Mechanics of Atoms
Chapter 28 Quantum Mechanics of Atoms 28.1 Quantum Mechanics The Theory Quantum mechanics incorporates wave-particle duality, and successfully explains energy states in complex atoms and molecules, the
More informationquantization condition.
/8/016 PHYS 34 Modern Physics Atom II: Hydrogen Atom Roadmap for Exploring Hydrogen Atom Today Contents: a) Schrodinger Equation for Hydrogen Atom b) Angular Momentum in Quantum Mechanics c) Quantum Number
More informationDEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS
DEVIL PHYSICS THE BADDEST CLASS ON CAMPUS IB PHYSICS TSOKOS LESSON 1-1B: THE INTERACTION OF MATTER WITH RADIATION Introductory Video Quantum Mechanics Essential Idea: The microscopic quantum world offers
More informationPhysics 342 Lecture 23. Radial Separation. Lecture 23. Physics 342 Quantum Mechanics I
Physics 342 Lecture 23 Radial Separation Lecture 23 Physics 342 Quantum Mechanics I Friday, March 26th, 2010 We begin our spherical solutions with the simplest possible case zero potential. Aside from
More informationPhysics 1C Lecture 29B
Physics 1C Lecture 29B Emission Spectra! The easiest gas to analyze is hydrogen gas.! Four prominent visible lines were observed, as well as several ultraviolet lines.! In 1885, Johann Balmer, found a
More informationApplied Statistical Mechanics Lecture Note - 3 Quantum Mechanics Applications and Atomic Structures
Applied Statistical Mechanics Lecture Note - 3 Quantum Mechanics Applications and Atomic Structures Jeong Won Kang Department of Chemical Engineering Korea University Subjects Three Basic Types of Motions
More informationPhysics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom
Physics 228 Today: Ch 41: 1-3: 3D quantum mechanics, hydrogen atom Website: Sakai 01:750:228 or www.physics.rutgers.edu/ugrad/228 Happy April Fools Day Example / Worked Problems What is the ratio of the
More informationWe also find the development of famous Schrodinger equation to describe the quantization of energy levels of atoms.
Lecture 4 TITLE: Quantization of radiation and matter: Wave-Particle duality Objectives In this lecture, we will discuss the development of quantization of matter and light. We will understand the need
More informationPhysics 43 Chapter 41 Homework #11 Key
Physics 43 Chapter 4 Homework # Key π sin. A particle in an infinitely deep square well has a wave function given by ( ) for and zero otherwise. Determine the epectation value of. Determine the probability
More informationChapter 38. Photons and Matter Waves
Chapter 38 Photons and Matter Waves The sub-atomic world behaves very differently from the world of our ordinary experiences. Quantum physics deals with this strange world and has successfully answered
More informationElectron Arrangement - Part 1
Brad Collins Electron Arrangement - Part 1 Chapter 8 Some images Copyright The McGraw-Hill Companies, Inc. Properties of Waves Wavelength (λ) is the distance between identical points on successive waves.
More informationCHAPTER 5 Wave Properties of Matter and Quantum Mechanics I
CHAPTER 5 Wave Properties of Matter and Quantum Mechanics I 5.1 X-Ray Scattering 5.2 De Broglie Waves 5.3 Electron Scattering 5.4 Wave Motion 5.5 Waves or Particles? 5.6 Uncertainty Principle 5.7 Probability,
More informationUnderstand the basic principles of spectroscopy using selection rules and the energy levels. Derive Hund s Rule from the symmetrization postulate.
CHEM 5314: Advanced Physical Chemistry Overall Goals: Use quantum mechanics to understand that molecules have quantized translational, rotational, vibrational, and electronic energy levels. In a large
More information