Ch 9 pages ; Lecture 21 – Schrodinger’s equation

Wave Equations - The Classical Wave Equation In classical mechanics, wave functions are obtained by solving a differential equation. For example, the one- dimensional wave equation for a vibrating string with linear mass  (units of kg/m) and tension  (units of force) is: The wave velocity is given by c and the wave function quantifies the vertical displacement of the string as a function of x and t

Wave Equations - The Classical Wave Equation Any function of the form: is a solution of the wave equation, where the specific forms for the wave frequency n and the wavelength n are determined by the details of the problem For example, for a harmonically vibrating string, fixed at x=0 and x=L (i.e. with boundary conditions The frequencies n are the harmonics of the vibrating string.

Wave Equations - The Classical Wave Equation Any linear combination of wave functions A wave function that is independent of time is called a standing wave. The wave equation for a standing wave is: (where c n are constant) is also a solution to the wave equation.

Wave Equations - The Classical Wave Equation Where  is a constant. If the boundary conditions are then any function of the form: is a solution if

Standing waves

Wave functions and experimental observables Particle wave functions are obtained by solving a quantum mechanical wave equation, called the Schroedinger equation In classical mechanics, the solution to the wave equation  (x,t)  describes the displacement (e.g. of a string) as a function of time and place In quantum mechanics, Schrodinger and Heisenberg introduced an analogous concept called wave function  (x,t)

Schroedinger’s quantum mechanical wave equation Schrodinger introduced his famous equation to calculate the value of the wave function for a particle in a potential V(x,t), the time-dependent Schroedinger equation is: If the potential V is independent of time, the wave function has the simple form: Where C is a constant and  satisfies the time-independent Schroedinger’s equation which has a simpler form:

Schroedinger’s quantum mechanical wave equation Solutions of this equation are independent of time, and are called stationary or standing particle waves, in analogy to classical standing waves in a vibrating string. Schroedinger’s equation for a particle in an infinitely deep well is: It is identical in form to the classical equation for a standing wave.

Wave functions and experimental observables Its physical interpretation is not immediate The square of the wave unction  (x,t) 2 characterizes the electron distribution in space and is a measure of the probability of finding an electron (or any other particle) at a certain time and place For example, the probability of finding a particle within a certain volume in space is given by:

Wave functions and experimental observables Although the wave function is a mathematical concept, it is of fundamental importance and can be directly measured (sort of) For example, X-ray diffraction experiments directly measure (apart from a Fourier transform) the square of the electron distribution of the material The chemical bond can only be described and understood by calculating wave functions for the electron in a molecule and so on.

Electron density from x-ray crystallography

The Hamiltonian: The Wavefunction: E = energy Describes a system in a given state The Hamiltonian is an operator What do we actually measure? Operators

Position xmultiply by x Momentump x Kinetic energyk x Potential energyV(x)multiply by V(x) Operators are associated with observables

Wave functions and experimental observables We can calculate the energy of a particle from its wave function and any other property of the system A fundamental tenet of quantum mechanics is that observables can be derived once the wave function is known. However, the duality of matter introduced earlier introduces a probabilistic nature to measurements, so that we can calculate observables only in a probabilistic sense

Wave functions and experimental observables This is done through the expectation value of a variable O, which can be calculated using the expression: Given a function of a complex variable f=a+ib, the complex conjugate f*=a-ib For example, the average position of an electron in a molecule is given by:

time-independent Schroedinger equation: particle in a box Using the Schroedinger Equation we can obtain the energies and wave functions for a particle in a box. Particle-in-a-Box refers to a particle of mass m in a potential defined as: The wave function has the form Where  (x) is obtained by solving the time-independent Schroedinger equation: With the requirement that to reflect the boundary conditions imposed by the potential V(x).

time-independent Schroedinger equation: particle in a box Which can be rearranged to a familiar form Since the particle must remain in the box where V(x)=0, the Schroedinger equation simplifies to: The general solution to this equation is:

time-independent Schroedinger equation: particle in a box Can only be satisfied if A=0 and However, the boundary conditions (the particle cannot leave the box!) The analogy with standing waves (vibrating strings) is well worth noting. Since, by definition: n=0, 1, 2, 3, … The quantized energy is obtained by substituting the expression for the wave function into Schrodinger’s equation and solving for the energy.

time-independent Schroedinger equation: particle in a box It is found to be: We can use the result to calculate the probability that the particle in state n is at position x: The lowest energy level (n=1) is called ground state, the others are called excited states. These are very important concepts in spectroscopy. To determine the constants A n we can recall that all probabilities must sum to one because the particle must be somewhere in the box

time-independent Schroedinger equation: particle in a box Which can be rearranged to give: Final answer:

What does the energy look like? Energy is quantized  E n = 1, 2, … Wavefunctions for particle in the box

Consider the following dye molecule, the length of which can be considered the length of the “box” an electron is limited to: What wavelength of light corresponds to  E from n=1 to n=2? L = 8 Å (experimental: 680 nm) Application/Example

Solving the Quantum Mechanical Wave Equation If the potential is independent of time i.e. V=V(x), the Schroedinger equation can be solved as follows. 1. Assume the wave function is a product of a function dependent only on x and a function only dependent on t: 2. Substitute that expression into the Schroedinger equation: 3. Divide both sides of the equation by Because the left-hand-side of the equation is dependent only on x, and the right-hand-side is dependent only on t, both sides must equal a constant. It can be shown to be the energy E.

Solving the Quantum Mechanical Wave Equation 4. The time equation: This equation has the general solution: 5. The space-dependent equation: is called the stationary or time independent Schroedinger equation. The solution  (x) depends on the potential V(x) and the boundary conditions imposed Can be re-written as:

Solving the Quantum Mechanical Wave Equation To summarize, the general solution to the Schroedinger equation (if the potential V(x) is independent of time): is Where  (x) is obtained by solving the time-independent Schroedinger equation:

Particle in a 3D box The Schroedinger equation for a particle in a three- dimensional box with dimensions a, b, c is: This equation can be solved exactly as for a one-dimensional case by assuming: As before The time-independent wave equation is:

Particle in a 3D box This equation can be further separated into three identical equations of the form of the one-dimensional particle-in-a- box equation. The result is that the energy is a sum of three identical terms: The wave function is a product of the form: