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.

Photons: Light Waves behaving as Particles

Photons:  Electromagnetic radiation behaves as both waves and particles.  The energy in an electromagnetic wave is carried in units called photons.  The energy E of one photon is proportional to the wave frequency f and inversely proportional to the wavelength lambda, and is proportional to a universal quantity h called Planck’s constant: E=hf=frac{hc}{lambda}.  The momentum of a photon has magnitude E/c: p=frac Ec=frac{hf}c=frac h{lambda}.

The photo-electric effect:  In the photo-electric effect, a surface can eject an electron by absorbing a photon whose energy hf is greater than or equal to the work function phi of the material.  The stopping potential V_0 is the voltage required to stop a current of ejected electrons from reaching an anode: eV_0=hf-phi.

Photon production, photon scattering, and pair production:  X rays can be produced when electrons accelerated to high kinetic energy across a potential increase V_{AC} strike a target.  The photon model explains why the maximum frequency and minimum wavelength produced are given by the equation: eV_{AC}=hf_{max}=frac{hc}{lambda_{min}} (bremsstrahlung).  In Compton scattering a photon transfers some of its energy and momentum to an electron with which it collides.  For free electrons (mass m), the wavelengths of incident and scattered photons are related to the photon scattering angle phi: lambda'-lambda=frac{h}{mc}(1-cosphi) (Compton scattering).  In pair production a photon of sufficient energy can disappear and be replaced by electron-positron pair.  In the inverse process, an electron and positron can annihilate and be replaced by a pair of photons.

The Heisenberg uncertainty principle:  It is impossible to determine both a photon’s position and its momentum at the same time to arbitrarily high precision.  The precision of such measurements for the x-components is limited by the Heisenberg uncertainty principle, Delta xDelta p_xgeqhbar/2; there are corresponding relationships for the y– and z-components.  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.  In these expressions, hbar=h/2pi.


Invariance of physical laws, simultaneity:  All of the fundamental laws of physics have the same form in all inertial frames of reference.  The speed of light in vacuum is the same in all inertial frames and is independent of the motion of the source.  Simultaneity is not an absolute concept; events that are simultaneous in one frame are not necessarily simultaneous in a second frame moving relative to the first.

Time dilation:  If two events occur at the same space point in a particular frame of reference, the time interval Delta t_0 between the events as measured in that frame is called a proper time interval.  If this frame moves with constant velocity u relative to a second frame, the time interval Delta t between the events as observed in the second frame is longer than Delta t_0: Delta t=frac{Delta t_0}{sqrt{1-frac{u^2}{c^2}}}=gammaDelta t_0, gamma=frac1{sqrt{1-u^2/c^2}}.

Length contraction:  If two points are at rest in a particular frame of reference, the distance l_0 between the points as measured in that frame is called a proper length. If this frame moves with constant velocity u relative to a second frame and the distances are measured parallel to the motion, the distance l between the points as measured in the second frame is shorter than l_0.  l=l_0sqrt{1-frac{u^2}{c^2}}=frac{l_0}{gamma}.

The Lorentz transformation:  The Lorentz coordinate transformations relate the coordinates and time of an event in an inertial frame S to the coordinates and time of the same event as observed in a second inertial frame S' moving at velocity u relative to the first.  For one-dimensional motion, a particle’s velocities v_x in S and v_x' in S' are related by the Lorentz velocity transformation.  x'=frac{x-ut}{sqrt{1-u^2/c^2}}=gamma(x-ut), y'=y, z'=z, t'=frac{t-ux/c^2}{sqrt{1-u^2/c^2}}=gamma(t-ux/c^2), v_x'=frac{v_x-u}{1-uv_x/c^2}, v_x=frac{v_x'+u}{1+uv_x'/c^2}.

The Doppler effect for electromagnetic waves:  The Doppler effect is the frequency shift in light from a source due to the relative motion of source and observer.  For a source moving toward the observer with speed u, the received frequency f in terms of the emitted frequency f_0 is f=sqrt{frac{c+u}{c-u}}f_0.

Relativistic momentum and energy:  For a particle of rest mass m moving with velocity vec{v}, the relativistic momentum is vec{p}=frac{mvec{v}}{sqrt{1-v^2/c^2}}=gamma mvec{v}, the relativistic kinetic energy is K=frac{mc^2}{sqrt{1-v^2/c^2}}-mc^2=(gamma-1)mc^2.  The total energy E is the sum of the kinetic energy and the rest energy mc^2: E=K+mc^2=frac{mc^2}{sqrt{1-v^2/c^2}}=gamma mc^2.  Also, E^2=(mc^2)^2+(pc)^2.


Fresnel and Fraunhofer diffraction:  Diffraction occurs when light passes through an aperture or around an edge.  When the source and the observer are so far away from the obstructing surface that the outgoing rays can be considered parallel, it is called Fraunhofer diffraction.  When the source or the observer is relatively close to the obstructive surface, it is Fresnel diffraction.

Single-slit diffraction:  Monochromatic light sent through a narrow slit of width a produces a diffraction pattern on a distant screen.  The condition for destructive interference (a dark fringe) at a point in the pattern at angle theta: sintheta=frac{mlambda}{a}, (m=pm1,pm2,pm3,ldots).  The intensity in the pattern as a function of theta: I=I_0{frac{sin[pi a(sintheta)/lambda]}{pi a(sintheta)/lambda}}^2.

Diffraction gratings:  A diffraction grating consists of a large number of thin parallel slits, spaced a distance d apart.  The condition for maximum intensity in the interference pattern is the same as for the two-source pattern, but the maxima for the grating are very sharp and narrow.  dsintheta=mlambda (m=pm1,pm2,pm3,ldots).

X-ray diffraction:  A crystal serves as a three-dimensional diffraction grating for x rays with wavelengths of the same order of magnitude as the spacing between atoms in the crystal.  For a set of crystal planes spaced a distance d apart, constructive interference occurs when the angles of incidence and scattering (measured from the crystal planes) are equal and when the Bragg condition is satisfied: 2dsintheta=mlambda, (m=1,2,3,ldots).

Circular apertures and resolving power:  The diffraction pattern from a circular aperture of diameter D consists of a central bright spot, called the Airy disk, and a series of concentric dark and bright rings.  Equation sintheta_1=1.22frac{lambda}D gives the angular radius theta_1 of the first dark ring, equal to the angular size of the Airy disk.  Diffraction sets the ultimate limit on resolution (image sharpness) of optical instruments.  According to Rayleigh’s criterion, two point objects are just barely resolved when their angular separation theta is given by the equation.


Interference and coherent sources:  Monochromatic light is light with a single frequency.  Coherence is a definite, unchanging phase relationship between two waves.  The overlap of waves from two coherent sources of monochromatic light forms an interference pattern.  The principle of superposition states that the total wave disturbance at any point is the sum of the disturbances from the separate waves.

Two-source interference of light:  When two sources are in phase, constructive interference occurs where the difference in path length from the two sources is zero or an integer number of wavelengths; destructive interference occurs where path difference is a half-integer number of wavelengths.  If two sources separated by a distance d are both very far from a point P, and the line from the sources make an angle theta with the line perpendicular to the line of the sources, then the condition for constructive interference at P is dsintheta=mlambda, (m=0,pm 1,pm 2,ldots).  The condition for destructive interference is dsintheta=(m+frac12)lambda, (m=0,pm1,pm2,ldots).  When theta is very small, the position y_m of the mth bright fringe on a screen located at distance R from the sources is: y_m=Rfrac{mlambda}d, (m=0,pm1,pm2,ldots).

Intensity in interference pattern:  When two sinusoidal waves with equal amplitude E and phase difference phi are superimposed, the resultant amplitude E_P and intensity I are as follows: E_P=2E|cosfrac{phi}2|, I=I_0cos^2frac{phi}2.  If the two sources emit in phase, the phase difference phi at a point P (located a distance r_1 from source 1 and a distance r_2 from source 2) is directly proportional to the path difference r_2-r_1: phi=frac{2pi}{lambda}(r_2-r_1)=k(r_2-r_1).

Interference in thin films:  When light is reflected from both sides of a thin film of thickness t and no phase shift occurs at either surface, constructive interference between the reflected waves occurs when 2t is equal to an integral number of wavelengths.  If a half-cycle phase shift occurs at one surface, this is the condition for destructive interference.  A half-cycle phase shift occurs during reflection whenever the index of refraction in the second material is greater than in the first.

Michelson interferometer:  The Michelson interferometer uses a monochromatic light source and can be used for high-precision measurements of wavelengths.  Its original purpose was to detect motion of the earth relative to a hypothetical ether, the supposed medium for electromagnetic waves.  The ether has never been detected, and the concept has been abandoned; the speed of light is the same relative to all observers.  This is part of the foundation of the special theory of relativity.