“Anyone who can contemplate quantum mechanics without getting dizzy hasn’t understood it ” Neils Bohr Discussion Is there a particle flying faster than light speed ? Lecture 10 Particles in 3D Potenti[.]
“Anyone who can contemplate quantum mechanics without getting dizzy hasn’t understood it.” Neils Bohr Discussion: Is there a particle flying faster than light speed ? Lecture 10: Particles in 3D Potentials and the Hydrogen Atom r / ao (r ) e ao ( x, y, z ) ( x) ( y) ( z ) z 11 L L P(r) g( x) 0.5 x L En x n y n z h 2 n n n x y z 8mL2 r = a0 00 0 r2 4a0 x En 13.6 eV n2 Content 3-Dimensional Potential Well Early Models of the Hydrogen Atom Product Wavefunctions Concept of degeneracy Planetary Model Quantum Modifications Schrödinger’s Equation for the Hydrogen Atom Ground state solution Spherically-symmetric excited states (“s-states”) Quantum Particles in 3D Potentials So far, we have considered quantum particles bound in one-dimensional potentials This situation can be applicable to certain physical systems but it lacks some of the features of many “real” 3D quantum systems, such as atoms and artificial quantum structures: A real (3D) “quantum dot” (www.kfa-juelich.de/isi/) One consequence of confining a quantum particle in two or three dimensions is “degeneracy” the occurrence of several quantum states at the same energy level To illustrate this important point in a simple system, we extend our favorite potential the infinite square well -to three dimensions Particle in a 3D Box (1) The extension of the Schrödinger Equation (SEQ) to 3D is straightforward in cartesian (x,y,z) coordinates: d d d U ( x , y , z ) E 2m dx dy dz like p x p 2y p z2 2m where ( x, y, z ) Kinetic energy term in the Schrödinger Equation Let’s solve this SEQ for the particle in a 3D box: z outside box, x or y or z < U(x,y,z) = L y L L x inside box outside box, x or y or z > L This simple U(x,y,z) can be “separated” U(x,y,z) = U(x) + U(y) + U(z) Particle in a 3D Box (2) So the Schrödinger Equation becomes: d d d U ( x ) U ( y ) U ( z ) E 2m dx dy dz and the wavefunction ( x, y, z ) can be “separated” into the product of three functions: ( x, y, z ) ( x) ( y) ( z ) So, the whole problem simplifies into three one-dimensional equations that we’ve already solved in Lecture d ( x) U ( x) ( x) Ex ( x) 2m dx n n ( x) N sin x L h nx x Enx 2m L d ( y) U ( y) ( y) E y ( y) 2m dy n y n ( y) N sin L y Likewise for (z) h2 n y Eny 2m L graphic 2 Particle in a 3D Box (3) So, finally, the eigenstates and associated energies for a particle in a 3D box are: nx n y nz N sin x sin y sin L L L h2 2 En x n y n z n n n x y z 8mL z where nx,ny, and nz can each have values 1,2,3,… z L y x L L This problem illustrates important new points (1) Three ‘quantum numbers’ (nx,ny,nz) are needed to completely identify the state of this three-dimensional system (2) More than one state can have the same energy: “Degeneracy” Degeneracy reflects an underlying symmetry in U(x,y,z) equivalent directions Lecture 10, exercise Consider a particle in a two-dimensional (infinite) well, with Lx = Ly Compare the energies of the (2,2), (1,3), and (3,1) states? a E(2,2) > E(1,3) = E(3,1) b E(2,2) = E(1,3) = E(3,1) c E(1,3) = E(3,1) > E(2,2) If we squeeze the box in the x-direction (i.e., Lx < Ly) compare E(1,3) with E(3,1): a E(1,3) < E(3,1) b E(1,3) = E(3,1) c E(1,3) > E(3,1) Lecture 10, exercise Consider a particle in a two-dimensional (infinite) well, with Lx = Ly Compare the energies of the (2,2), (1,3), and (3,1) states? a E(2,2) > E(1,3) = E(3,1) b E(2,2) = E(1,3) = E(3,1) E(1,3) = E(1,3) = E0 (12 + 32) = 10 E0 c E(1,3) = E(3,1) > E(2,2) E(2,2) = E0 (22 + 22) = E0 If we squeeze the box in the x-direction (i.e., Lx < Ly) compare E(1,3) with E(3,1): The tighter confinement along x will increase a E(1,3) < E(3,1) the contribution to E The effect will be b E(1,3) = E(3,1) greatest on states with greatest nx: c E(1,3) > E(3,1) Example: Lx = Ly/2 E( nx ny ) Enx ny h nx n y 8m Lx Ly h2 2 n n E(1,3) 9; E(3,1) 36 x y 8mLy Energy levels (1) Now back to a 3D cubic box: z Show energies and label (nx,ny,nz) for the first 11 states of the particle in the 3D box, and write the degeneracy D for each allowed energy Use Eo= h2/8mL2 L E y x L L ... potential the infinite square well -to three dimensions Particle in a 3D Box (1) The extension of the Schrödinger Equation (SEQ) to 3D is straightforward in cartesian (x,y,z) coordinates: ... graphic 2 Particle in a 3D Box (3) So, finally, the eigenstates and associated energies for a particle in a 3D box are: nx n y nz N sin x sin y sin L L .. .Lecture 10: Particles in 3D Potentials and the Hydrogen Atom r / ao (r ) e ao ( x, y, z ) ( x) ( y) ( z ) z 11