Paul Avery PHZ4390 Aug. 19, 2015 Basics of Quantum Mechanics 1 Introduction: The Wavefunction ............................................................................................... 1 1.1 Wavefunction and Probability ............................................................................................ 2 1.2 Superposition of Wavefunctions ........................................................................................ 2 1.3 Wavefunction Evolution and Measurement ....................................................................... 2 1.4 Stationary states .................................................................................................................. 2 1.5 Eigenvalues and Eigenstates ............................................................................................... 3 2 Expectation Value ...................................................................................................................... 3 3 Schrodinger Equation Examples ............................................................................................... 4 3.1 Particle in a box with infinite walls: one dimension .......................................................... 4 3.2 Harmonic oscillator: one dimension ................................................................................... 6 3.3 Particle in a box with infinite walls: three dimensions ....................................................... 7 3.4 The hydrogen atom: using spherical coordinates ............................................................... 9 4 The Uncertainty Principle in Quantum Mechanics ................................................................. 13 4.1 Uncertainty principle and ground state energy ................................................................. 14 4.2 Finding the approximate size of an atom .......................................................................... 14 5 Appendix: More Details of the Hydrogen Atom Solution ...................................................... 15 References ..................................................................................................................................... 17 1 Introduction: The Wavefunction The state of a particle is described by a complex continuous wave function ψ q of some coor- ( ) dinates q. In this brief summary the coordinates q are typically chosen to be (x,t), and other coor- dinates can be added for a more complete description, e.g., the particle’s total spin and spin pro- jection. Examples of wavefunctions include • ψ (x) = 2 / L sin(nπ x / L) (n integer) describes a particle confined to a one-dimensional box in the interval [0, L]. • ψ x,t = ei(p⋅x−iE)t/ / V describes a free particle in a box of volume V with momentum ( ) p, including time dependence. r/a • ψ r,θ,φ = 1/ π a3e− 0 , where a is the Bohr radius, describes the ground state of a ( ) 0 0 hydrogen atom. Wavefunctions are normalized so that ψ * x,t ψ x,t dx = 1 ∫ ( ) ( ) i.e., total probability = 1. The connection between wavefunction and probability is explored in the next subsection. 1 1.1 Wavefunction and Probability 2 dP = ψ q dq represents the probability of finding a particle with coordinates q in the interval ( ) 2 dq (assuming that q is a continuous variable, like coordinate x or momentum p). Thus ψ q is ( ) a probability density. In the free particle example above, the probability for the particle having x,y,z > 0 is P = ψ (x,t) d 3x = d 3x / V = 1 / 8 . For the particle in the one dimensional box, ∫ΔV ∫ΔV the probability of the particle in its ground state (n = 1) being found in the first third of the box is L/3 P = 2 / L sin2 π x / L dx = 0.1956 . Only probabilities and observables (see below) can be ( ) ∫0 ( ) measured, not ψ itself. 1.2 Superposition of Wavefunctions If ψ q describes some physical state 1 and ψ q describes a physical state 2, then the wave- 1( ) 2 ( ) function ψ q = αψ q + βψ q describes a new physical state that is called the superposition ( ) 1( ) 2 ( ) of the two states 1 and 2. If the states are normalized and orthogonal (“orthonormal”), then 2 2 α + β = 1. 1.3 Wavefunction Evolution and Measurement The notion of an experiment (observation) is a key concept in understanding quantum mechan- ics: only by conducting an experiment one can localize a particle with probabilities calculated from ψ q . Before an experiment is conducted, the particle cannot be said to be at a given coor- ( ) dinate. But neither does the wavefunction describe some kind of smeared distribution of matter. QM describes the deterministic evolution of ψ q , which gives probabilistic predictions for ex- ( ) periment outcomes. For example, quantum mechanics can predict the angular distribution of electrons scattered through a narrow slit (the angular distribution is a probability distribution), but cannot determine the trajectory of any given electron. The equation describing the (nonrelativistic) time evolution of ψ is called the Schrodinger equa- tion, after the physicist who first described it. It can be written as ∂ i ψ (q) = Hˆψ (q) ∂t ˆ ˆ 2 where H is a hamiltonian operator, which in most cases is H = pˆ / 2m +U ., where pˆ is the momentum operator and U is the potential. It has been empirically shown that the QM probabil- istic description cannot be reduced to some deterministic, but hidden parameters.1 1.4 Stationary states Most physical potentials are time independent, so the Hamiltonian operator has no explicit time dependence. This suggests that we use separation of variables and write the wavefunction as the product of a spatial part and a time part ψ x,t =ψ x T t . Substituting this in the Schrodinger ( ) ( ) ( ) equation and dividing by ψT gives 2 T′ Hˆψ −i = T ψ The left and right sides of the equation are functions of different variables, thus each must be constant. We call the constant E (it has units of energy) and obtain ψ x,t =ψ x e−iEt/ , where ( ) ( ) ψ x , the non-time part of the wavefunction, is a “stationary state”. Thus all time independent ( ) potentials reduce to the simplified Schrodinger equation Hˆψ x = Eψ x ( ) ( ) 1.5 Eigenvalues and Eigenstates Consider a QM operator gˆ . If a wavefunction ψ q exists such that g ( ) gˆψ q = gψ q g ( ) g ( ) where g is an ordinary (complex) number, we say that ψ q is an eigenstate of gˆ with eigen- g ( ) value g. The following examples should help explain the idea. • In the time-independent Schrodinger equation, the time dependent part of the wavefunc- tion ψ = e−iEt/ is an eigenstate of the time evolution operator Tˆ = i ∂ with eigenvalue t ∂t E (energy): Tˆψ = Eψ . t t The free particle wavefunction eipx/ is an eigenstate of the momentum operator • ψ = pˆ = −i ∂ with eigenvalue p (momentum): pˆ eipx/ = peipx/ . x ∂x x • The wavefunction eimφ is an eigenstate of the z angular momentum operator Lˆ = −i ∂ z ∂φ im im with eigenvalue m ( m is z component of angular momentum): Lˆ e φ = me φ . z 2 Expectation Value Consider a QM operator gˆ . For any wavefunction ψ q the expectation value of gˆ for that ( ) wavefunction is defined as ψ gˆ ψ ≡ ψ ∗ q gˆψ q dq ∫ ( ) ( ) 2 Since ψ q dq is the probability density, the expectation value can be considered to be the ( ) ∗ usual statistical notion of expectation value. However, the order ψ , gˆ , ψ is important because gˆ is an operator and typically operates to the right, as is the case for differential operators. Note that if the wavefunction is an eigenstate of gˆ with eigenvalue g, then the expectation value is merely g. 3 Let’s calculate the momentum expectation value for a particle in a one dimensional box with in- finite potential walls (Section 3). The wavefunction is ψ x = 2 / L sin nπ x / L . The expecta- ( ) ( ) tion value is 2 ⎛ nπ x⎞ ⎛ d ⎞ ⎛ nπ x⎞ ψ pˆ ψ = sin −i sin dx = 0 L ∫ ⎝⎜ L ⎠⎟ ⎝⎜ dx ⎠⎟ ⎝⎜ L ⎠⎟ ikx −ikx A value of zero is expected because sin kx ~ e − e , which is an equal combination of a par- ticle moving to the right and a particle moving to the left with the same momentum. We can also work out the expectation of the momentum squared operator 2 ⎛ nπ x⎞ ⎛ d 2 ⎞ ⎛ nπ x⎞ n2π 22 ψ pˆ 2 ψ = sin −2 sin dx = ∫ ⎜ ⎟ ⎜ 2 ⎟ ⎜ ⎟ 2 L ⎝ L ⎠ ⎝ dx ⎠ ⎝ L ⎠ L This is nonzero because the wavefunction has separate components moving to the left and the right. 3 Schrodinger Equation Examples 3.1 Particle in a box with infinite walls: one dimension The particle in a 1-D box neatly illustrates how quantization arises when a potential is present. The idea is a particle confined to a region of length L, which we accomplish with the following potential: U (x) = 0 0 ≤ x ≤ L U x = ∞ Otherwise ( ) Classically, a particle within the region can have any positive energy, bouncing back and forth between the walls with a momentum depending on the energy. In the QM picture, we write the stationary state Schrodinger equation inside the region as 2 ∂2 − ψ x = Eψ x 2m 2 ( ) ( ) ∂x which can be rearranged into the form of a standard second order differential equation 2mE x x 0 ψ ′′( ) + 2 ψ ( ) = Although in principle the energy E can have any value, the value must allow the wavefunction to be 0 at the two boundaries of the region to maintain continuity with the vanishing wave function 2 2 outside the region. We first try negative energies, or 2mE = − k , for which the general solu- tion is ψ x = Aekx + Be−kx , where A and B are constants. However, the condition that ψ 0 = 0 ( ) ( ) and ψ L = 0 force A = B = 0 , so no nontrivial solution with negative energy exists. ( ) 4 We next try positive energies or 2mE = 2k 2 , with solution ψ x = Asin kx + Bcoskx . The ( ) boundary condition ψ 0 = 0 forces B = 0, while ψ L = 0 leads to sin kL 0 as the only non- ( ) ( ) = trivial solution. This can be satisfied only if kL is a multiple of π, e.g. k = nπ / L , for any non- zero integer n. The boundary conditions thus lead us to wavefunctions of the form 2 2 2 2 ψ n x = Asin nπ x / L with energy En = n π / 2mL . Normalizing to unit probability ( ) ( ) ∞ 2 ψ x dx = 1 gives A = 2 / L . Thus the only solutions to the Schrodinger equation of a par- ∫−∞ ( ) ticle in an infinitely high box are: 2 ⎛ nπ x⎞ iE t/ ψ x,t = sin e− n n ≥ 1 n ( ) L ⎝⎜ L ⎠⎟ n22π 2 E n = 2 2mL i.e., only a countable number of solutions and their corresponding energies are possible.