Lect11 - Anyone who can contemplate quantum mechanics...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon
“Anyone who can contemplate quantum mechanics without getting dizzy hasn’t understood it.” --Niels Bohr
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Particles in 3D Potentials and the Hydrogen Atom P(r) 0 4a 0 0 1 r r = a 0 z x L L L ( ) 2 2 2 2 2 8 z y x n n n n n n mL h E z y x + + = ) ( ) ( ) ( ) , , ( z y x z y x ϕ ψ = o a / r 3 o e a 1 ) r ( = π 2 6 13 n eV . E n =
Background image of page 2
Overview Overview z 3-Dimensional Potential Well z Product Wavefunctions z Concept of degeneracy z Early Models of the Hydrogen Atom z Planetary Model z Schrödinger’s Equation for the Hydrogen Atom z Semi-quantitative picture from uncertainty principle z Ground state solution* z Spherically-symmetric excited states (“s-states”)* z *contain details beyond what we expect you to absorb
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Quantum Particles in 3D Potentials Quantum Particles in 3D Potentials z 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. z 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: (www.kfa-juelich.de/isi/) A real (3D) “quantum dot” z To illustrate this important point in a simple system, we extend our favorite potential -- the infinite square well -- to three dimensions.
Background image of page 4
Particle in a 3D Box (1) Particle in a 3D Box (1) z The extension of the Schrödinger Equation (SEQ) to 3D is straightforward in cartesian (x,y,z) coordinates: ψ E ) z , y , x ( U dz d dy d dx d m = + + + 2 2 2 2 2 2 2 2 = ) , , ( z y x ψ≡ where () 2 2 2 2 1 z y x p p p m like + + Kinetic energy term in the Schrödinger Equation outside box, x or y or z < 0 outside box, x or y or z > L 0 inside box U(x,y,z) = z Let’s solve this SEQ for the particle in a 3D box: y This simple U(x,y,z) can be “separated” A special feature U(x,y,z) = U(x) + U(y) + U(z) z x L L L
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Particle in a 3D Box (2) Particle in a 3D Box (2) z So the Schrödinger Equation becomes: () ψ E ) z ( U ) y ( U ) x ( U dz d dy d dx d m = + + + + + 2 2 2 2 2 2 2 2 = and the wavefunction can be “separated” into the product of three functions: ) , , ( z y x ψ≡ ) ( ) ( ) ( ) , , ( z y x z y x ϕ = z So, the whole problem simplifies into three one-dimensional equations that we’ve already solved in Lecture 7. ) ( ) ( ) ( ) ( 2 2 2 2 x E x x U dx x d m x = + = ) ( ) ( ) ( ) ( 2 2 2 2 y E y y U dy y d m y = + = = x L n N x x n π sin ) ( 2 2 2 2 = L n m h E x nx = y L n N y y n sin ) ( 2 2 2 2 = L n m h E y ny ..Likewise for ϕ ( z ).. graphic
Background image of page 6
Particle in a 3D Box (3) Particle in a 3D Box (3) z So, finally, the eigenstates and associated energies for a particle in a 3D box are: z x y L L L = z L n y L n x L n N z y x π ψ sin sin sin ( ) 2 2 2 2 2 8 z y x n n n n n n mL h E z y x + + = where n x ,n y , and n z can each have values 1,2,3,….
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 8
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 09/15/2008 for the course PHYS 214 taught by Professor Debevec during the Spring '07 term at University of Illinois at Urbana–Champaign.

Page1 / 31

Lect11 - Anyone who can contemplate quantum mechanics...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online