Solutions of Schrodinger's Equation: Particle in a Box - Slides | PHYS 486, Study notes of Quantum Physics

Download Solutions of Schrodinger's Equation: Particle in a Box - Slides | PHYS 486 and more Study notes Quantum Physics in PDF only on Docsity! Page 1 Physics 486 Lecture 4 Physics 486, Spring ‘07 Lecture 4 Solutions of Schrödinger’s Equation: Particle in a Box The Schrödinger Equation Recall from last lecture that there are two different forms for the “matter” wave equation by Schrodinger: A time-independent Schrödinger Equation: 2 2 22 V E m x ψ ψ ψ− ∂ + = ∂ The time-dependent Schrödinger Equation: 2 2 22 i V t m x ψ ψ ψ∂ − ∂= + ∂ ∂ Most general form Special case Special Solutions to SEQ: Eigenstates To see the connection between these two forms of the SEQ, let’s consider the general time-dependent equation: This method is known as a “separation of variables”, and we’ll see that these solutions correspond to a special case of interest to us. Let’s consider solutions to this general equation that can be written as a product of functions: ( )txtx ϕψ )(),( =Ψ ( ) ( ) ( ) t txitxxV x tx m ∂ ∂ =+ ∂ ∂ − ,),(, 2 2 22 ψψψ The resulting time-dependent SEQ can be written: ( ) ( ) ( ) ( ) ( ) t txitxxV x tx m ∂ ∂ =+ ∂ ∂ − ϕψϕψϕψ )()( 2 2 22 Special Solutions to SEQ: Eigenstates Which can then be written: ( ) ( ) ( ) ( ) ( ) t txitxxV x xt m ∂ ∂ =+ ∂ ∂ − ϕψϕψψϕ )()( 2 2 22 Dividing both sides of the above equation by ( )txtx ϕψ )(),( =Ψ ( ) ( ) ( ) ( ) ( ) t t t ixV x x xm ∂ ∂ =+ ∂ ∂ − ϕ ϕ ψ ψ 11 2 2 22 Note that we now have an equation that depends only on x on the left, and an equation that depends only on t on the right. The only way that both equations can equal one another for all x and t is if they are equal to the same constant, C: : ( ) ( ) C t t t i = ∂ ∂ϕ ϕ 1 ⇒ ( ) ( )tC t ti ϕϕ = ∂ ∂ Page 2 Physics 486 Lecture 4 Special Solutions to SEQ: Eigenstates Also: Clearly, comparing the above result with our earlier result for the time-independent SEQ, C = total energy, E What does this mean? The solutions to this form of the SEQ (these are called eigenstates of the SEQ equation, for reasons we’ll discuss soon) are those having a single well-defined energy E! What is the time dependence of these “special” solutions. To see this, let’s solve for ϕ(t): ( ) ( ) ( ) CxV dx xd xm =+− 2 22 1 2 ψ ψ ( ) ( ) ( ) ( )xCxxV dx xd m ψψψ =+− 2 22 2⇒ ( ) ( ) ( ) ( )xExxV dx xd m ψψψ =+− 2 22 2 ( ) ( )tE dt tdi ϕϕ = ⇒ ( ) ( ) ( )titEi dt td ωϕϕϕ −=−= ( ) tiet ωϕ −=⇒ Time-dependence of Eigenstates So, solutions to the SEQ with a single well-defined energy E (“eigenstates”) have a time-dependence: Ee)x()t,x( ti =ωψ=Ψ ω− with What is the physical significance of this complex time dependence? Nothing because what we measure is |Ψ(x,t)|2: 2ti*ti2 )x(e)x(e)x()t,x( ψ=ψ×ψ=Ψ ω+ω− Therefore, the probability density |Ψ(x,t)|2 associated with eigenstates of the SEQ don’t change with time!! So, eigenstates of the SEQ, i.e., quantum states with a well-defined energy, are stationary states, in the sense that their probability densities don’t evolve with time! Thus, the SEQ provides a clear explanation for Bohr’s stationary orbit postulate! The “stationary orbits” are simply solutions to the time-independent SEQ! This wavefunction has a “complex” time-dependence The Time-Independent Schrödinger Equation )()()()( 2 2 22 xExxV dx xd m ψψψ =+− To reiterate, the probability densities |ψ | 2 associated with the solutions to the time-independent SEQ do not change with time…they are in “stationary states” For the next few lectures, we will focus on the time-independent SEQ for a variety of physically interesting situations. Again, these situations are pertinent ONLY when the particle’s wavefunction is associated with a single energy E (we’ll come back to the more general case later). For the time being, we’ll also focus on the 1-dimensional case: ψ(x,y,z) ψ(x) π = 2 h Application of SEQ: Particle in Box KE term PE term Total E term V = 0 for 0 < x < L V = ∞ everywhere else )()()()( 2 2 22 xExxV dx xd m ψψψ =+− V(x) 0 L ∞ ∞ As a specific important example, consider a quantum particle confined to a small region, 0 < x < L, by infinite potential walls. We call this a “one-dimensional (1D) box”. Recall, from last lecture, the time-independent SEQ in one dimension: Page 5 Physics 486 Lecture 4 Eigenstates of ISW Potential      = x L nBxn πψ sin)( We need to “normalize” this wavefunction, i.e., find the “normalization factor” B, by setting the total probability of finding the particle in the well equal to one: 2 2 2 2 0 Total probability sin 2 L n Ldx B x dx B L πψ    = = =       ∫ ∫ = 1 B = What about the allowed solutions (eigenstates) for the ISW potential? 1/ 22 L       n = 1,2,3,… So, the general solutions (“eigenstates”) for this potential are:     = x L n L xn πψ sin2)( n = 1,2,3,… V=∞ ψ(x) 0 L V=∞n=1 n=2 x n=3 Particles in Confined Potentials Important general result from the “particle-in-box”: Energy quantization in all “quantum” systems, including atoms, arises because only particle wavefunctions that “fit” within the constraints of the confining potential, and their associated energies, are allowed Quantum example: Particle-in-box Only wavefunctions (and corresponding energies) having an integral # of half-wavelengths that precisely “fit” in well are allowed E B Quantized energy levels in a real “quantum dot” (portal.research.bell-labs.com) Standing waves: classical example Only frequencies corresponding to wavelengths that “fit” within constraints give standing waves Standing waves on a constrained string Real standing waves of electron density in a “quantum corral” IBM Almaden Single atoms (Fe) Cu Standing waves: quantum example Page 6 Physics 486 Lecture 4 Particle-in-Box: Example Calculate ground state energy and energy for a transition. An electron is trapped in a “quantum nanobox” that is L = 4 nm long. Assuming that the potential seen by the electron is approximately that of an infinite square well, estimate the ground (lowest) state energy of the electron. Allowed energies for electron in a 1D box: What photon energy is required to excite the trapped electron to the next available energy level (i.e., n = 2)? L/n2 n =λ 1 2 n EnE = So, energy difference between n = 2 and n = 1 levels: ( ) 122 E 12E −=∆ 1E3= = 0.071 eV E1 = ground state energy (n = 1) V=∞ V=∞ 0 xL En n=1 n=2 n=3 eV 0235.0 )nm4(4 nmeV 505.1E L4 nmeV 505.1 mL8 hE withnEE nmeV 505.1 m2 hE 2 2 1 2 2 2 2 1 2 1n 2 n 2 2 n 2 n = ⋅ = ⋅ === ⋅ == λλ Probability Densities ψ V=∞ V=∞ 0 xL ψ 0 xL ψ 0 xL The square of the eigenstates give the probability densities, |ψ(x)|2:      = x L n L xn πψ sin2)(      = x L n L xn πψ 22 sin2)( n=1 |ψ|2 0 xL 2/L n=2 |ψ|2 0 xL 2/L |ψ|2 0 xL n=3 2/L General Properties of Bound States Several trends exhibited by the particle-in-box states are generic to bound state wavefunctions in any potential (even complicated ones). (2). The lowest energy bound state always has finite energy -- a “zero-point” energy Why? Even the lowest energy bound state requires some wavefunction curvature (kinetic energy) to satisfy boundary conditions (1) The overall curvature of the wavefunction increases with increasing kinetic energy, T = E-V. 2 22 )( 2 dx xd m ψ −       m2 plike 2 V=∞ ψ(x) 0 L V=∞n=1 n=2 x n=3 V=∞ V=∞ 0 xL En n=1 n=2 n=3 T1 T2 T3 General Properties of Bound States (3). The -th eigenstate) has ( -1) zero-crossings. (e.g., the first (ground) eigenstate has 0, the second eigenstate has 1, the third eigenstate has 2, etc.,…) (4). If the potential V(x) has a center of symmetry (such as the center of the well above), the eigenstates will be (alternating) even and odd functions about that center of symmetry V=∞ ψ(x) 0 L V=∞n=1 n=2 x n=3 Page 7 Physics 486 Lecture 4 General Properties of Bound States (5). Effects of a change in potential: A change in potential will be reflected in a change in the curvature of the wavefunction (see rule #1), since the curvature increases with increasing T=E-V(x). But what about the amplitude of the wavefunction? V=∞ 0 L V=∞ x I II Consider the change in potential between regions I and II above. The wavefunction, and its derivative, can be written: )sin( ϕψ += kxA )cos(/ ϕψ += kxAkdxd Squaring both of these relations and adding gives: ( ) ( ))(cos)(sin)(cos)(sin/ 2222222212 ϕϕϕϕψψ +++=+++=+ − kxkxAkxAkxAdxdk ( ) 2212 / Adxdk =+ − ψψ ( )212 / dxdkA ψψ −+= Across the boundary between I and II, both ψ and dψ/dx MUST be continuous, while k can change discontinuously. So, as k=(2m(E-V)/ 2)1/2 decreases across the boundary, A must increase. Classically, we would say that the probability amplitude increases w/ decreasing k because the particle moves more slowly there. Bound State Properties: Example E5 V=∞ V=∞ 0 L x Uo x ψ Let’s reinforce your intuition about the properties of bound state wavefunctions with this example: Through “nano-engineering,” you are able to create a “step” in the potential “seen” by an electron trapped in a 1D structure, as shown below. You’d like to estimate the wavefunction for an electron in the 5th energy level of this potential, without solving the SEQ. Qualitatively sketch the 5th wavefunction: Bound State Properties: Example E5 V=∞ V=∞ 0 L x Uo x ψ Let’s reinforce your intuition about the properties of bound state wavefunctions with this example: Through “nano-engineering,” you are able to create a “step” in the potential “seen” by an electron trapped in a 1D structure, as shown below. You’d like to estimate the wavefunction for an electron in the 5th energy level of this potential, without solving the SEQ. Qualitatively sketch the 5th wavefunction: Things to consider: (2) Wavefunction must go to zero at x = 0 and x = L. (3) Kinetic energy is lower on right side of well, so the curvature of ψ is smaller there (wavelength is longer). (4) Kinetic energy is lower on right side, so amplitude of ψ is larger there. (Classically, the particle spends more time there because it is moving more slowly). (1) 5th wavefunction has 4 zero-crossings The wavefunction below describes a quantum particle in a range ∆x: ψ(x) x ∆x In what energy level is the particle? n = (a) 7 (b) 8 (c) 9 Bound States: Another Example