Momentum in Quantum Mechanics

For a particle in state Psi, the expectation value of x is langle xrangle=int_{-infty}^{+infty}x|Psi(x,t)|^2,dx.

langle prangle=mfrac{dlangle xrangle}{dt}=-ihbarint_{-infty}^{+infty}left(Psi^*frac{partialPsi}{partial x}right),dx.

In general, langle Q(x,p)rangle=int_{-infty}^{+infty}Psi^*,Qleft(x,frac{hbar}{i}frac{partial}{partial x}right)Psi,dx.

For example, T=frac12mv^2=frac{p^2}{2m}, so langle Trangle=-frac{hbar^2}{2m}int_{-infty}^{+infty}Psi^*frac{partial^2Psi}{partial x^2},dx.


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^2Psi(x,t)}{partial x^2}+U(x)Psi(x,t)=ihbarfrac{partialPsi(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^2psi(x)}{dx^2}+U(x)psi(x)=Epsi(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^2pi^2hbar^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}sinfrac{npi 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 dpsi(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 dpsi(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)hbarsqrt{frac{k'}{m}}=(n+frac12)hbaromega (n=1,2,3,ldots).  The spacing between any two adjacent levels is hbaromega, 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.

Particles Behaving as Waves

De Broglie waves and electron diffraction:  Electrons and other particles have wave properties.  A particle’s wavelength depends on its momentum in the same way as for photons: lambda=frac hp=frac h{mv}, E=hf.  A non-relativistic electron accelerated from rest through a potential difference V_{ba} has a wavelength lambda=frac hp=frac h{sqrt{2meV_{ba}}}.  Electron microscopes use the very small wavelengths of fast-moving electrons to make images with resolution thousands of times finer than is possible with visible light.

The nuclear atom:  The Rutherford scattering experiments show that most of an atom’s mass and all of its positive charge are concentrated in a tiny, dense nucleus at the center of the atom.

Atomic line spectra and energy levels:  The energies of atoms are quantized: They can have only certain definite values, called energy levels.  When an atom makes a transition from an energy level E_i to a lower level E_f, it emits a photon of energy E_i-E_f: hf=frac{hc}{lambda}=E_i-E_f.  The same photon can be absorbed by an atom in the lower energy level, which excites the atom to the upper level.

The Bohr model:  In the Bohr model of the hydrogen atom, the permitted values of angular momentum are integral multiples of h/2pi: L_n=mv_nr_n=nfrac{h}{2pi}, (n=1,2,3,ldots).  The integer multiplier n is called the principal quantum number for the level.  The orbital radii are proportional to n^2: r_n=epsilon_0frac{n^2h^2}{pi me^2}=n^2a_0, v_n=frac{1}{epsilon_0}frac{e^2}{2nh}.  The energy levels of the hydrogen atoms are given by E_n=-frac{hcR}{n^2}=-frac{13.60,mathrm{eV}}{n^2}, (n=1,2,3,ldots), where R is the Rydberg constant.

The laser:  The laser operates on the principle of stimulated emission, by which many photons with identical wavelength and phase are emitted.  Laser operation requires a nonequilibrium condition called population inversion, in which more atoms are in a higher-energy state than are in a lower-energy state.

Blackbody radiation:  The total radiated intensity (average power radiated per area) from a blackbody surface is proportional to the fourth power of the absolute temperature T: I=sigma T^4 (Stefan-Boltzmann law).  The quantity sigma=5.67times 10^{-8},mathrm{W/m^2cdot K^4} is called the Stefan-Boltzmann constant.  The wavelength lambda_m at which a blackbody radiates most strongly is inversely proportional to T: lambda_mT=2.90times 10^{-3},mathrm{mcdot K} (Wien displacement law).  The Planck radiation law gives the spectral emittance I(lambda) (intensity per wavelength interval in blackbody radiation): I(lambda)=frac{2pi hc^2}{lambda^5(e^{hc/lambda kT}-1)}.

The Heisenberg uncertainty principle for particles:  The same uncertainty considerations that apply to photons also apply to particles such as electrons.  The uncertainty Delta E in the energy of a state that is occupied for a time Delta t is given by equation Delta tDelta Egeqhbar/2.