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Chapter 40 Quantum Mechanics Using Schrödinger Equation • • • • • • To study the “particle in a box” To consider and construct wave functions To study a finite potential well To examine quantum mechanical behavior around a barrier and “tunneling” To consider the harmonic oscillator—our first model for molecular vibrations To study three-dimensional systems Erwin Schrödinger 1887 – 1961 American physicist Best known as one of the creators of quantum mechanics His approach was shown to be equivalent to Heisenberg’s Also worked with: statistical mechanics color vision general relativity Schrödinger Equation The Schrödinger equation as it applies to a particle of mass m confined to moving along the x axis and interacting with its environment through a potential energy function U(x) is h 2 d 2ψ − + Uψ = Eψ 2 2m dx This is called the time-independent Schrödinger equation Schrödinger Equation, cont. For most cases, the first term in the Schrödinger equation reduces to the kinetic energy of the particle multiplied by the wave function Solutions to the Schrödinger equation in different regions must join smoothly at the boundaries Schrödinger Equation, final ψ(x) must be continuous dψ/dx must also be continuous for finite values of the potential energy Solutions of the Schrödinger Equation Solutions of the Schrödinger equation may be very difficult The Schrödinger equation has been extremely successful in explaining the behavior of atomic and nuclear systems Classical physics failed to explain this behavior When quantum mechanics is applied to macroscopic objects, the results agree with classical physics Wave Function The complete wave function ψ for a system depends on the positions of all the particles in the system and on time The function can be written as r r r r r − iωt Ψ = ( r1 + r2 + r3 + K + rj K,t ) = Ψ ( rj ) e rj is the position of the jth particle in the system ω = 2πƒ is the angular frequency Wave Function, cont. The wave function is often complex-valued The absolute square |ψ|2 = ψ*ψ is always real and positive ψ* is the complete conjugate of ψ It is proportional to the probability per unit volume of finding a particle at a given point at some instant The wave function contains within it all the information that can be known about the particle Wave Function Interpretation – Single Particle Ψ cannot be measured |Ψ|2 is real and can be measured |Ψ|2 is also called the probability density The relative probability per unit volume that the particle will be found at any given point in the volume If dV is a small volume element surrounding some point, the probability of finding the particle in that volume element is P(x, y, z) dV = |Ψ |2 dV Wave Function of a Free Particle The wave function of a free particle moving along the x-axis can be written as ψ(x) = Aeikx A is the constant amplitude k = 2π/λ is the angular wave number of the wave representing the particle Although the wave function is often associated with the particle, it is more properly determined by the particle and its interaction with its environment Think of the system wave function instead of the particle wave function Wave Function of a Free Particle, cont. In general, the probability of finding the particle in a volume dV is |ψ|2 dV With one - dimensional analysis, this becomes |ψ|2 dx The probability of finding the particle in the arbitrary interval a ≤ x ≤ b is b 2 Pab = ∫ ψ dx a and is the area under the curve Wave Function of a Free Particle, Final Because the particle must be somewhere along the x axis, the sum of all the probabilities over all values of x must be 1 ∞ 2 Pab = ∫ ψ dx = 1 −∞ Any wave function satisfying this equation is said to be normalized Normalization is simply a statement that the particle exists at some point in space Expectation Values Measurable quantities of a particle can be derived from ψ Remember, ψ is not a measurable quantity Once the wave function is known, it is possible to calculate the average position you would expect to find the particle after many measurements The average position is called the expectation value of x and is defined as ∞ x ≡ ∫ ψ * xψdx −∞ Expectation Values, cont. The expectation value of any function of x can also be found ∞ f ( x ) = ∫ ψ * f ( x ) ψdx −∞ The expectation values are analogous to weighted averages Such as velocity, momentum, rootmean-square displacement…etc Summary of Mathematical Features of a Wave Function ψ(x) may be a complex function or a real function, depending on the system ψ(x) must be defined at all points in space and be single-valued ψ(x) must be normalized ψ(x) must be continuous in space There must be no discontinuous jumps in the value of the wave function at any point Particle in a Box A particle is confined to a one-dimensional region of space The “box” is onedimensional The particle is bouncing elastically back and forth between two impenetrable walls separated by L Potential Energy for a Particle in a Box As long as the particle is inside the box, the potential energy does not depend on its location We can choose this energy value to be zero The energy is infinitely large if the particle is outside the box This ensures that the wave function is zero outside the box Wave Function for the Particle in a Box Since the walls are impenetrable, there is zero probability of finding the particle outside the box ψ(x) = 0 for x < 0 and x > L The wave function must also be 0 at the walls The function must be continuous ψ(0) = 0 and ψ(L) = 0 Schrödinger Equation Applied to a Particle in a Box In the region 0 < x < L, where U = 0, the Schrödinger equation can be expressed in the form d 2ψ 2mE 2mE 2 = − 2 ψ = −k ψ where k = 2 dx h h The most general solution to the equation is ψ(x) = A sin kx + B cos kx A and B are constants determined by the boundary and normalization conditions Particle in a Box (cont’d) Apply boundary conditions to ψ(x) = A sin kx + B cos kx At x = 0, Ψ(0)= 0 + B = 0 → B=0 At x = L, Ψ(L)= A sin(kL) = 0 → k = nπ/L Math – particle in a box Ψ(x) = A sin(kx), where k = nπ/L Normalization required 1= ∞ L −∞ 0 2 2 * dx A sin ( kx ) dx Ψ Ψ = ∫ ∫ L L 1 L ∫0 sin (kx )dx = ∫0 2[1 − cos(2kx )]dx = 2 2 So A = √(2/L) and Ψ ( x ) = 2 nπ sin( x) L L Graphical Representations for a Particle in a Box Wave Function of the Particle in a Box, cont. Only certain wavelengths for the particle are allowed |ψ|2 is zero at the boundaries |ψ|2 is zero at other locations as well, depending on the values of n The number of zero points increases by one each time the quantum number increases by one Momentum of the Particle in a Box Remember the wavelengths are restricted to specific values λ= 2 L / n Therefore, the momentum values are also restricted h nh p= = λ 2L Energy of a Particle in a Box We chose the potential energy of the particle to be zero inside the box Therefore, the energy of the particle is just its kinetic energy (E = ћ2k2/2m, k=nπ/L) h2 2 En = n 2 8mL n = 1, 2, 3 ,K The energy of the particle is quantized Energy Level Diagram – Particle in a Box The lowest allowed energy corresponds to the ground state En = n2E1 are called excited states E = 0 is not an allowed state The particle can never be at rest Boundary Conditions Boundary conditions are applied to determine the allowed states of the system In the model of a particle under boundary conditions, an interaction of a particle with its environment represents one or more boundary conditions and, if the interaction restricts the particle to a finite region of space, results in quantization of the energy of the system In general, boundary conditions are related to the coordinates describing the problem Potential Wells A potential well is a graphical representation of energy The well is the upward-facing region of the curve in a potential energy diagram The particle in a box is sometimes said to be in a square well Due to the shape of the potential energy diagram Finite Potential Well A finite potential well is pictured The energy is zero when the particle is 0<x<L In region II The energy has a finite value outside this region Regions I and III Classical vs. Quantum Interpretation According to Classical Mechanics If the total energy E of the system is less than U, the particle is permanently bound in the potential well If the particle were outside the well, its kinetic energy would be negative (Impossible!!) According to Quantum Mechanics A finite probability exists that the particle can be found outside the well even if E < U The uncertainty principle allows the particle to be outside the well as long as the apparent violation of conservation of energy does not exist in any measurable way Finite Potential Well – Region II U=0 The allowed wave functions are sinusoidal The boundary conditions no longer require that ψ be zero at the ends of the well The general solution will be ψII(x) = F sin kx + G cos kx where F and G are constants Finite Potential Well – Regions I and III The Schrödinger equation for these regions may be written as d 2ψ 2m (U − E ) = ψ 2 2 dx h The general solution of this equation is ψ = AeCx + Be −Cx A and B are constants Finite Potential Well – Regions I and III, cont. In region I, B = 0 In region III, A = 0 This is necessary to avoid an infinite value for ψ for large negative values of x This is necessary to avoid an infinite value for ψ for large positive values of x The solutions of the wave equation become Cx −Cx Ψ I = Ae for x < 0 and Ψ III = Be for x > L Finite Potential Well – Graphical Results for ψ The wave functions for various states are shown Outside the potential well, classical physics forbids the presence of the particle Quantum mechanics shows the wave function decays exponentially to approach zero Finite Potential Well – Graphical Results for ψ2 The probability densities for the lowest three states are shown The functions are smooth at the boundaries Finite Potential Well – Determining the Constants The constants in the equations can be determined by the boundary conditions and the normalization condition The boundary conditions are dψI dψII ψI = ψII and at x = 0 = dx dx dψII dψIII ψII = ψIII and = at x = L dx dx Usually a lengthy calculation… Boundary Conditions ψI(x)=Aecx ψII(x)=F sinkx + ψIII(x)=Be-cx ψI(0)= ψII(0) ψI’(0)= ψII’(0) ψII(L)= ψIII(L) ψII’(L)= ψIII’(L) G coskx A=G cA=kF FsinkL+GcoskL=Be-cx k(FcoskL-GsinkL)=-cBe-cx Transcendental Equation k(F – G tankL)/(F tankL + G) = -c with c/k = F/A ≡ α (α – tankL)/(α tankL + 1) = -α It can be simplified to tan kL = -2α/(α2 – 1) again a=c/k; k2=2mE/ћ2, c2=2m(Uo-E)/ћ2 There is no analytical solution, need to solve numerically. The above equation is often referred to as a “transcendental equation”. Solutions For Uo=6E∞ where E∞ =h2/8mL2 Total E≡bE∞, α2=(6-b)/b, kL=π√b 20 RHS tan 15 -2α/(α2-1) Only 3 solutions possible 10 tankL 5 0 -5 0 1 2 3 4 5 -10 -15 -20 0.625 2.43 b (E=bEinf) 5.09 6 E1=0.625E∞ E2=2.43E∞ E3=5.09E∞ Eigenfunctions and Eigenvalues Application – Nanotechnology Nanotechnology refers to the design and application of devices having dimensions ranging from 1 to 100 nm Nanotechnology uses the idea of trapping particles in potential wells One area of nanotechnology of interest to researchers is the quantum dot A quantum dot is a small region that is grown in a silicon crystal that acts as a potential well Potential barriers and tunneling A fun novel to learn about tunneling is Mr. Thompkins in Wonderland by Gamov. In the book, after attending a lecture on quantum theory, Mr. Thompkins arrives home and parks in his garage. He awakens to find his car out in the driveway! Tunneling The potential energy has a constant value U in the region of width L and zero in all other regions This a called a square barrier U is the called the barrier height Tunneling, cont. Classically, the particle is reflected by the barrier Regions II and III would be forbidden According to quantum mechanics, all regions are accessible to the particle The probability of the particle being in a classically forbidden region is low, but not zero According to the uncertainty principle, the particle can be inside the barrier as long as the time interval is short and consistent with the principle Tunneling, final The curve in the diagram represents a full solution to the Schrödinger equation Movement of the particle to the far side of the barrier is called tunneling or barrier penetration The probability of tunneling can be described with a transmission coefficient, T, and a reflection coefficient, R Tunneling Coefficients The transmission coefficient represents the probability (ψ*ψ) that the particle penetrates to the other side of the barrier The reflection coefficient represents the probability that the particle is reflected by the barrier T+R=1 The particle must be either transmitted or reflected T ≈ e-2CL and can be nonzero Tunneling is observed and provides evidence of the principles of quantum mechanics Applications of Tunneling Alpha decay In order for the alpha particle to escape from the nucleus, it must penetrate a barrier whose energy is several times greater than the energy of the nucleus-alpha particle system Nuclear fusion Protons can tunnel through the barrier caused by their mutual electrostatic repulsion Alpha Decay Potential Barrier Examples of tunnelling Tunnelling occurs in many situations in physics and astronomy: 1. Nuclear fusion (in stars and fusion reactors) V Strong nuclear force (attractive) Coulomb interaction (repulsive) Incident particles 2. Alpha-decay V Internuclear distance x ( Ze) 2 Barrier height ~ ~ MeV 4πε 0 rnucleus thermal energies (~keV) Initial αparticle energy Distance x of αparticle from nucleus More Applications of Tunneling – Scanning Tunneling Microscope An electrically conducting probe with a very sharp edge is brought near the surface to be studied The empty space between the tip and the surface represents the “barrier” The tip and the surface are two walls of the “potential well” Scanning Tunneling Microscope The STM allows highly detailed images of surfaces with resolutions comparable to the size of a single atom At right is the surface of graphite “viewed” with the STM Scanning Tunneling Microscope, final The STM is very sensitive to the distance from the tip to the surface This is the thickness of the barrier STM has one very serious limitation Its operation is dependent on the electrical conductivity of the sample and the tip Most materials are not electrically conductive at their surfaces The atomic force microscope overcomes this limitation Real atoms in a 2-D well Following our model, surface Scientists placed 48 Fe atoms on a Cu surface (this is done at ultra-high vacuum and the image is made with scanning-tunneling electron microscopy). Tunneling Microscope Simple Harmonic Oscillator Reconsider black body radiation as vibrating charges acting as simple harmonic oscillator Important because many systems can be approximated by SHO (small oscillations) The potential energy is U = ½ kx2 = ½ mω2x2 Its total energy is E = K + U = ½ kA2 = ½ mω2A2 SHO – ground state The Schrödinger equation for this problem is h 2 d 2ψ 1 2 2 − + mω x ψ = Eψ 2 2m dx 2 The ground state solution of this equation is − ( mω 2 h ) x 2 B is a constant ψ = Be where c = mω/2ћ ψ’ = -2cxBe−cx2 ψ’’ = -2Be-cx2 + 4c2x2Be-cx2 = -2cψ + 4c2x2ψ So the Schrödinger equation becomes -2cψ + 4c2x2ψ - 4c2x2ψ = −(2mE/ћ2)ψ or E = cћ2/m = ½ ћω (ground state energy) General Solution - SHO The remaining solutions that describe the excited states all include the exponential function 2 ψn(x) = Bn(x)e-Cx The energy levels of the oscillator are quantized. Solution of Bn(x) is related to the Hermite Polynomials The energy for an arbitrary quantum number n is En = (n + ½)ћω where n = 0, 1, 2,… The exact wave function is ψ n (u ) = where c x 2 u= 1 −u 2 / 2 H n ( u )e n 1/ 2 n!2 π Hermite Differential Equation y’’ – 2xy’ + 2ny = 0 Hn(x) = (-1)nex2dn/dxn(e-x2) Hn+1(x) = 2xHn(x) – 2nHn-1(x) H0(x) H1(x) H2(x) H3(x) H4(x) H5(x) = = = = = = 1 2x -2 + 4x2 -12x + 8x3 12 – 48x2 + 16x4 120x – 160x3 + 32x5 Energy Level Diagrams – Simple Harmonic Oscillator The separation between adjacent levels are equal and equal to ∆E = ћω The energy levels are equally spaced The state n = 0 corresponds to the ground state The energy is Eo = ½ ћω Agrees with Planck’s original equations Comparison of Newtonian and Quantum Oscillators Quantum mechanics in 3-D • • • The atomic/molecular world is a 3-D place. Cartesian coordinates work but are computationally cumbersome. The system that works best with spherical coordinates. But “2 becomes complicated. Question The graph below represents a wave function for a particle confined to . The value of the normalization constant A may be A –2 a. b. c. d. e. -1/4 -1/2 1/2 1/4 either -1/2 or +1/2 +2 x Question The graph below represents a wave function for a particle confined to . The value of the normalization constant A may be A –2 a. b. c. d. e. -1/4 -1/2 1/2 1/4 either -1/2 or +1/2 +2 x Question The first five wave functions for a particle in a box are shown. The probability of finding the particle near x = L/2 is A. least for n = 1. B. least for n = 2 and n = 4. C. least for n = 5. D. the same (and nonzero) for n = 1, 2, 3, 4, and 5. E. zero for n = 1, 2, 3, 4, and 5. Question The first five wave functions for a particle in a box are shown. The probability of finding the particle near x = L/2 is A. least for n = 1. B. least for n = 2 and n = 4. C. least for n = 5. D. the same (and nonzero) for n = 1, 2, 3, 4, and 5. E. zero for n = 1, 2, 3, 4, and 5. Question The first five wave functions for a particle in a box are shown. Compared to the n = 1 wave function, the n = 5 wave function has A. the same kinetic energy (KE). B. 5 times more KE. C. 25 times more KE. D. 125 times more KE. E. none of the above Question The first five wave functions for a particle in a box are shown. Compared to the n = 1 wave function, the n = 5 wave function has A. the same kinetic energy (KE). B. 5 times more KE. C. 25 times more KE. D. 125 times more KE. E. none of the above Question The first three wave functions for a finite square well are shown. The probability of finding the particle at x > L is A. least for n = 1. B. least for n = 2. C. least for n = 3. D. the same (and nonzero) for n = 1, 2, and 3. E. zero for n = 1, 2, and 3. Question The first three wave functions for a finite square well are shown. The probability of finding the particle at x > L is A. least for n = 1. B. least for n = 2. C. least for n = 3. D. the same (and nonzero) for n = 1, 2, and 3. E. zero for n = 1, 2, and 3. Question A potential-energy function is shown. If a quantum-mechanical particle has energy E < U0, it is impossible to find the particle in the region A. x < 0. B. 0 < x < L. C. x > L. D. misleading question — the particle can be found at any x Question A potential-energy function is shown. If a quantum-mechanical particle has energy E < U0, it is impossible to find the particle in the region A. x < 0. B. 0 < x < L. C. x > L. D. misleading question — the particle can be found at any x Question The figure shows the first six energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics. B. do not have a definite wavelength. C. are all equal to zero at x = 0. D. Both A. and B. are true. E. All of A., B., and C. are true. Question The figure shows the first six energy levels of a quantum-mechanical harmonic oscillator. The corresponding wave functions A. are nonzero outside the region allowed by Newtonian mechanics. B. do not have a definite wavelength. C. are all equal to zero at x = 0. D. Both A. and B. are true. (why?) E. All of A., B., and C. are true. Question θ θ(0) = 5o release from rest θ(t) = ? What does the differential equation looks like? 1 kg 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: Material A (e.g. AlGaAs) Material B (e.g. GaAs) Electron potential energy Position Now used in many electronic devices (some transistors, diodes, solid-state lasers) Kroemer Esaki