ExampleThe speed of anelectronis measuredto be 1000 m/sto an accuracy of 0.001%.Find theuncertaintyinthepositionofthiselectron.p = mv = (9.11 × 10-31 kg) (1 × 103 m/s)= 9.11 × 10-28 kg.m/sAp = p × 0.001% = 9.11 × 10-33 kg m/s△x= h / △p = 6.626 × 10-34 / (9.11 x 10-33)= 7.27 × 10-2 (m)The speed of a bullet of mass of 0.01 kg is measured to be 1000 m/s to an accuracy of0.001%.Findtheuncertaintyinthepositionofthisbulletp = mv = (0.01 kg) (1 × 103 m/s) = 10 kg.m/sAp=p × 0.001%= 1 × 10-4 kg m/s△x= h / △p = 6.626 × 10-34 / (1 × 10-4)= 6.626 × 10-30 (m)11
Example The speed of an electron is measured to be 1000 m/s to an accuracy of 0.001%. Find the uncertainty in the position of this electron. p = mv = (9.11 10-31 kg) (1 103 m/s) = 9.11 10-28 kg.m/s p = p 0.001% = 9.11 10-33 kg m/s x = h / p = 6.626 10-34 / (9.11 x 10-33 ) = 7.27 10-2 (m) The speed of a bullet of mass of 0.01 kg is measured to be 1000 m/s to an accuracy of 0.001%. Find the uncertainty in the position of this bullet. p = mv = (0.01 kg) (1 103 m/s) = 10 kg.m/s p = p 0.001% = 1 10-4 kg m/s x = h / p = 6.626 10-34 / (1 10-4 ) = 6.626 10-30 (m) 11
P.DiracN.BohrM.PlanckW.Heisenberg(1902-1984)（1885-1962)(1858-1947)(1901-1976)E.Schrodinger(1887-1961)A.Einstein(1878-1955)M.BornW.PauliL. de Broglie(1882-1970)(1900-1958)(1892-1987)12
M. Planck (1858-1947) A. Einstein (1878-1955) L. de Broglie (1892-1987) P. Dirac (1902-1984) E. Schrödinger (1887-1961) W. Heisenberg (1901-1976) N. Bohr (1885-1962) W. Pauli (1900-1958) M. Born (1882-1970) 12
① color②greenhouse effect③electricresistance13
① color ② greenhouse effect ③ electric resistance 13
1.3WavefunctionsA quantum mechanical system is described by its wavefunction, which is a function of thepositions of all the particles in the system. The symbol y or is commonly used for thewavefunction.Example: We can approximate the vibrational motion of a diatomic molecule by a harmonicoscillator with effective mass m, force constant k, and position x. For the lowest-energy statethewavefunctioniskn(1)V。=Nexp2hwhereh is Planck's constant h divided by 2元, and N is a constant. This wavefunction isreal,butingeneralwavefunctionsmaybecomplexWeshallseelaterhowtoobtainthiswavefunction14
1.3 Wavefunctions A quantum mechanical system is described by its wavefunction, which is a function of the positions of all the particles in the system. The symbol ψ or Ψ is commonly used for the wavefunction. Example: We can approximate the vibrational motion of a diatomic molecule by a harmonic oscillator with effective mass m, force constant k, and position x. For the lowest-energy state the wavefunction is 2 0= exp 2 km N x where ℏ is Planck’s constant ℎ divided by 2π, and N is a constant. This wavefunction is real, but in general wavefunctions may be complex. We shall see later how to obtain this wavefunction. (1) 14
The wavefunction is a mathematical map of the system: it contains within itself informationabouteverything that any experimenton the system canpossibly measure.Wehaveto domathsonthewavefunctiontoextractthisinformation.Thesimplestinformationtoobtainistheprobabilityoffindingthesystemataparticularposition,thisisproportionaltothesquare of the wavefunction (or to its square modulus if complex).For the harmonic oscillator ground-state wavefunction (1) this givesP(x)dx ocyoyodxP(x)ky= N? expdxhX+dXfor the probability of finding the particle between x and x + dxP(x)is calledtheprobabilitydensity15
The wavefunction is a mathematical map of the system: it contains within itself information about everything that any experiment on the system can possibly measure. We have to do maths on the wave function to extract this information. The simplest information to obtain is the probability of finding the system at a particular position; this is proportional to the square of the wavefunction (or to its square modulus if complex). For the harmonic oscillator ground-state wavefunction (1) this gives * 0 0 2 2 d d exp d P x x x km N x x for the probability of finding the particle between x and x + dx. P(x) is called the probability density 15