{"id":209,"date":"2019-04-09T01:05:33","date_gmt":"2019-04-09T05:05:33","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/?post_type=chapter&p=209"},"modified":"2019-04-10T14:37:56","modified_gmt":"2019-04-10T18:37:56","slug":"chapter-3-review","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/bcitphys8400\/chapter\/chapter-3-review\/","title":{"raw":"Chapter 3 Review","rendered":"Chapter 3 Review"},"content":{"raw":"
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Key Terms<\/span><\/h3>\r\n
\r\n \t
anti-symmetric function<\/dt>\r\n \t
odd function<\/dd>\r\n<\/dl>\r\n
\r\n \t
Born interpretation<\/dt>\r\n \t
states that the square of a wave function is the probability density<\/dd>\r\n<\/dl>\r\n
\r\n \t
complex function<\/dt>\r\n \t
function containing both real and imaginary parts<\/dd>\r\n<\/dl>\r\n
\r\n \t
Copenhagen interpretation<\/dt>\r\n \t
states that when an observer\u00a0<\/span>is not<\/em>\u00a0<\/span>looking or when a measurement is not being made, the particle has many values of measurable quantities, such as position<\/dd>\r\n<\/dl>\r\n
\r\n \t
correspondence principle<\/dt>\r\n \t
in the limit of large energies, the predictions of quantum mechanics agree with the predictions of classical mechanics<\/dd>\r\n<\/dl>\r\n
\r\n \t
energy levels<\/dt>\r\n \t
states of definite energy, often represented by horizontal lines in an energy \u201cladder\u201d diagram<\/dd>\r\n<\/dl>\r\n
\r\n \t
energy quantum number<\/dt>\r\n \t
index that labels the allowed energy states<\/dd>\r\n<\/dl>\r\n
\r\n \t
energy-time uncertainty principle<\/dt>\r\n \t
energy-time relation for uncertainties in the simultaneous measurements of the energy of a quantum state and of its lifetime<\/dd>\r\n<\/dl>\r\n
\r\n \t
even function<\/dt>\r\n \t
in one dimension, a function symmetric with the origin of the coordinate system<\/dd>\r\n<\/dl>\r\n
\r\n \t
expectation value<\/dt>\r\n \t
average value of the physical quantity assuming a large number of particles with the same wave function<\/dd>\r\n<\/dl>\r\n
\r\n \t
field emission<\/dt>\r\n \t
electron emission from conductor surfaces when a strong external electric field is applied in normal direction to conductor\u2019s surface<\/dd>\r\n<\/dl>\r\n
\r\n \t
ground state energy<\/dt>\r\n \t
lowest energy state in the energy spectrum<\/dd>\r\n<\/dl>\r\n
\r\n \t
Heisenberg\u2019s uncertainty principle<\/dt>\r\n \t
places limits on what can be known from a simultaneous measurements of position and momentum; states that if the uncertainty on position is small then the uncertainty on momentum is large, and vice versa<\/dd>\r\n<\/dl>\r\n
\r\n \t
infinite square well<\/dt>\r\n \t
potential function that is zero in a fixed range and infinitely beyond this range<\/dd>\r\n<\/dl>\r\n
\r\n \t
momentum operator<\/dt>\r\n \t
operator that corresponds to the momentum of a particle<\/dd>\r\n<\/dl>\r\n
\r\n \t
nanotechnology<\/dt>\r\n \t
technology that is based on manipulation of nanostructures such as molecules or individual atoms to produce nano-devices such as integrated circuits<\/dd>\r\n<\/dl>\r\n
\r\n \t
normalization condition<\/dt>\r\n \t
requires that the probability density integrated over the entire physical space results in the number one<\/dd>\r\n<\/dl>\r\n
\r\n \t
odd function<\/dt>\r\n \t
in one dimension, a function antisymmetric with the origin of the coordinate system<\/dd>\r\n<\/dl>\r\n
\r\n \t
position operator<\/dt>\r\n \t
operator that corresponds to the position of a particle<\/dd>\r\n<\/dl>\r\n
\r\n \t
potential barrier<\/dt>\r\n \t
potential function that rises and falls with increasing values of position<\/dd>\r\n<\/dl>\r\n
\r\n \t
principal quantum number<\/dt>\r\n \t
energy quantum number<\/dd>\r\n<\/dl>\r\n
\r\n \t
probability density<\/dt>\r\n \t
square of the particle\u2019s wave function<\/dd>\r\n<\/dl>\r\n
\r\n \t
quantum dot<\/dt>\r\n \t
small region of a semiconductor nanocrystal embedded in another semiconductor nanocrystal, acting as a potential well for electrons<\/dd>\r\n<\/dl>\r\n
\r\n \t
quantum tunneling<\/dt>\r\n \t
phenomenon where particles penetrate through a potential energy barrier with a height greater than the total energy of the particles<\/dd>\r\n<\/dl>\r\n
\r\n \t
resonant tunneling<\/dt>\r\n \t
tunneling of electrons through a finite-height potential well that occurs only when electron energies match an energy level in the well, occurs in quantum dots<\/dd>\r\n<\/dl>\r\n
\r\n \t
resonant-tunneling diode<\/dt>\r\n \t
quantum dot with an applied voltage bias across it<\/dd>\r\n<\/dl>\r\n
\r\n \t
scanning tunneling microscope (STM)<\/dt>\r\n \t
device that utilizes quantum-tunneling phenomenon at metallic surfaces to obtain images of nanoscale structures<\/dd>\r\n<\/dl>\r\n
\r\n \t
Schr\u04e7dinger\u2019s time-dependent equation<\/dt>\r\n \t
equation in space and time that allows us to determine wave functions of a quantum particle<\/dd>\r\n<\/dl>\r\n
\r\n \t
Schr\u04e7dinger\u2019s time-independent equation<\/dt>\r\n \t
equation in space that allows us to determine wave functions of a quantum particle; this wave function must be multiplied by a time-modulation factor to obtain the time-dependent wave function<\/dd>\r\n<\/dl>\r\n
\r\n \t
standing wave state<\/dt>\r\n \t
stationary state for which the real and imaginary parts of\u00a0<\/span>\u03a8(x,t)<\/span><\/span><\/span>\u00a0<\/span>oscillate up and down like a standing wave (often modeled with sine and cosine functions)<\/dd>\r\n<\/dl>\r\n
\r\n \t
state reduction<\/dt>\r\n \t
hypothetical process in which an observed or detected particle \u201cjumps into\u201d a definite state, often described in terms of the collapse of the particle\u2019s wave function<\/dd>\r\n<\/dl>\r\n
\r\n \t
stationary state<\/dt>\r\n \t
state for which the probability density function,\u00a0<\/span>|\u03a8(x,t)|2<\/span><\/span><\/span>, does not vary in time<\/dd>\r\n<\/dl>\r\n
\r\n \t
time-modulation factor<\/dt>\r\n \t
factor\u00a0<\/span>e\u2212i\u03c9t<\/span><\/span><\/span>\u00a0<\/span>that multiplies the time-independent wave function when the potential energy of the particle is time independent<\/dd>\r\n<\/dl>\r\n
\r\n \t
transmission probability<\/dt>\r\n \t
also called tunneling probability, the probability that a particle will tunnel through a potential barrier<\/dd>\r\n<\/dl>\r\n
\r\n \t
tunnel diode<\/dt>\r\n \t
electron tunneling-junction between two different semiconductors<\/dd>\r\n<\/dl>\r\n
\r\n \t
tunneling probability<\/dt>\r\n \t
also called transmission probability, the probability that a particle will tunnel through a potential barrier<\/dd>\r\n<\/dl>\r\n
\r\n \t
wave function<\/dt>\r\n \t
function that represents the quantum state of a particle (quantum system)<\/dd>\r\n<\/dl>\r\n
\r\n \t
wave function collapse<\/dt>\r\n \t
equivalent to state reduction<\/dd>\r\n<\/dl>\r\n
\r\n \t
wave packet<\/dt>\r\n \t
superposition of many plane matter waves that can be used to represent a localized particle<\/dd>\r\n<\/dl>\r\n<\/div>\r\n
\r\n

Key Equations<\/span><\/h3>\r\n
\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Normalization condition in one dimension<\/td>\r\nP(x=\u2212\u221e,+\u221e)=\u222b\u2212\u221e\u221e|\u03a8(x,t)|2dx=1<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Probability of finding a particle in a narrow interval of position in one dimension\u00a0<\/span>(x,x+dx)<\/span><\/span><\/span><\/td>\r\nP(x,x+dx)=\u03a8*(x,t)\u03a8(x,t)dx<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Expectation value of position in one dimension<\/td>\r\n\u2329x\u232a=\u222b\u2212\u221e\u221e\u03a8*(x,t)x\u03a8(x,t)dx<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Heisenberg\u2019s position-momentum uncertainty principle<\/td>\r\n\u0394x\u0394p\u2265\u210f2<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Heisenberg\u2019s energy-time uncertainty principle<\/td>\r\n\u0394E\u0394t\u2265\u210f2<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Schr\u04e7dinger\u2019s time-dependent equation<\/td>\r\n\u2212\u210f22m\u22022\u03a8(x,t)\u2202x2+U(x,t)\u03a8(x,t)=i\u210f\u22022\u03a8(x,t)\u2202t<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
General form of the wave function for a time-independent potential in one dimension<\/td>\r\n\u03a8(x,t)=\u03c8(x)e\u2212i\u03c9t<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Schr\u04e7dinger\u2019s time-independent equation<\/td>\r\n\u2212\u210f22md2\u03c8(x)dx2+U(x)\u03c8(x)=E\u03c8(x)<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Schr\u04e7dinger\u2019s equation (free particle)<\/td>\r\n\u2212\u210f22m\u22022\u03c8(x)\u2202x2=E\u03c8(x)<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Allowed energies (particle in box of length\u00a0<\/span>L<\/em>)<\/td>\r\nEn=n2\u03c02\u210f22mL2,n=1,2,3,...<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Stationary states (particle in a box of length\u00a0<\/span>L<\/em>)<\/td>\r\n\u03c8n(x)=2Lsinn\u03c0xL,n=1,2,3,...<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Potential-energy function of a harmonic oscillator<\/td>\r\nU(x)=12m\u03c92x2<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Stationary Schr\u04e7dinger equation<\/td>\r\n\u2212\u210f2md2\u03c8(x)dx2+12m\u03c92x2\u03c8(x)=E\u03c8(x)<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
The energy spectrum<\/td>\r\nEn=(n+12)\u210f\u03c9,n=0,1,2,3,...<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
The energy wave functions<\/td>\r\n\u03c8n(x)=Nne\u2212\u03b22x2\/2Hn(\u03b2x),n=0,1,2,3,...<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Potential barrier<\/td>\r\nU(x)={0,whenx<0U0,when0\u2264x\u2264L0,whenx>L<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Definition of the transmission coefficient<\/td>\r\nT(L,E)=|\u03c8tra(x)|2|\u03c8in(x)|2<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
A parameter in the transmission coefficient<\/td>\r\n\u03b22=2m\u210f2(U0\u2212E)<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Transmission coefficient, exact<\/td>\r\nT(L,E)=1cosh2\u03b2L+(\u03b3\/2)2sinh2\u03b2L<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n
Transmission coefficient, approximate<\/td>\r\nT(L,E)=16EU0(1\u2212EU0)e\u22122\u03b2L<\/span><\/span><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/section><\/div>\r\n
\r\n

Summary<\/span><\/h3>\r\n
\r\n
\r\n

3.1<\/span>\u00a0<\/span><\/span>Wave Functions<\/span><\/a><\/h4>\r\n