Quantum Mechanics I: Wave Functions

Wave functions:  The wave function for a particle contains all of the information about that particle.  If the particle moves in one dimension in the presence of a potential energy function U(x), the wave function \Psi(x,t) obeys the one-dimensional Schrödinger equation: -\frac{\hbar^2}{2m}\frac{\partial^2\Psi(x,t)}{\partial x^2}+U(x)\Psi(x,t)=i\hbar\frac{\partial\Psi(x,t)}{\partial t}.  (For a free particle on which no forces act, U(x)=0.)  The quantity |\Psi(x,t)|^2, called the probability distribution function, determines the relative probability of finding a particle near a given position at a given time.  If the particle is in a state of definite energy, called a stationary state, \Psi(x,t) is a product of a function \psi(x) that depends on only spatial coordinates and a function e^{-iEt/\hbar} that depends on only time: \Psi(x,t)=\psi(x)e^{iEt/\hbar}.  For a stationary state, the probability distribution function is independent of time.

A spatial stationary-state wave function \psi(x) for a particle that moves in one dimension in the presence of a potential-energy function U(x) satisfies the time-independent Schrödinger equation: -\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2}+U(x)\psi(x)=E\psi(x).  More complex wave functions can be constructed by super-imposing stationary-state wave functions.  These can represent particles that are localized in a certain region, thus representing both particle and wave aspects.

Particle in a box:  The energy levels for a particle of mass m in a box (an infinitely deep square potential well) with width L are given by the equation: E_n=\frac{p_n^2}{2m}=\frac{n^2h^2}{8mL^2}=\frac{n^2\pi^2\hbar^2}{2mL^2} (n=1,2,3,\ldots).  The corresponding normalized stationary-state wave functions of the particle are given by the equation \psi_n(x)=\sqrt{\frac2L}\sin\frac{n\pi x}L (n=1,2,3,\ldots).

Wave functions and normalization:  To be a solution of the Schrödinger equation, the wave function \psi(x) and its derivative d\psi(x)/dx must be continuous everywhere except where the potential-energy function U(x) has an infinity discontinuity.  Wave functions are usually normalized so that the total probability of finding the particle somewhere is unity: \int_{-\infty}^{+\infty}|\psi(x)|^2\,dx=1.

Finite potential well:  In a potential well with finite depth U_0, the energy levels are lower than those for an infinitely deep well with the same width, and the number of energy levels corresponding to bound states is finite.  The levels are obtained by matching wave functions at the well walls to satisfy the continuity of \psi(x) and d\psi(x)/dx.

Potential barriers and tunneling:  There is a certain probability that a particle will penetrate a potential-energy barrier even though its initial energy is less than the barrier height.  This process is called tunneling.

Quantum harmonic oscillator:  The energy levels for the harmonic oscillator (for which U(x)=\frac12k'x^2) are given by the equation: E_n=(n+\frac12)\hbar\sqrt{\frac{k'}{m}}=(n+\frac12)\hbar\omega (n=1,2,3,\ldots).  The spacing between any two adjacent levels is \hbar\omega, where \omega=\sqrt{k'/m} is the oscillation angular frequency of the corresponding Newtonian harmonic oscillator.

Measurement in quantum mechanics:  If the wave function of a particle does not correspond to a definite value of a certain physical property (such as momentum or energy), the wave function changes when we measure that property.  This phenomenon is called wave-function collapse.


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