1 §27.1 The Heisenberg uncertainty principles Nature is bilateral: particles are waves and waves are particles.The particle aspect carries with it the traditional concepts of position and momentum; The wave aspect carries with it the concepts of wavelength and frequency. Nature places natural limits on the precision of our measurements; some knowledge and information forever is shrouded from our prying eyes. 2 1. The position-momentum uncertainty principle Single slit diffraction of electrons： . . . . . . . . λθ = 1 sina z p h a = 1 θ hpa z = 1 θ §27.1 The Heisenberg uncertainty principles 1 θ p r y z The position uncertainty： ay =? hpy y =??? 1 θ zyy ppp ==? p p z p y 1 θ The momentum uncertainty in y direction: 1 θ p r y §27.1 The Heisenberg uncertainty principles z If we account for the secondary maxima hpy y ≥???hpx x ≥??? hpz z ≥??? 3 The Heisenberg uncertainty principle states that there exists a fundamental limit to the extent to which we can simultaneously determine the position and corresponding momentum component of a wave-particle in any given direction. 2. The energy-time uncertainty principle )2 2 cos( )cos(),( txA tkxAtx πν λ π ωΨ ?= ?= Recall mathematical form of a monochromatic wave §27.1 The Heisenberg uncertainty principles νλ hE p h x == , Substitution for λ and ν ) 22 cos(),( Et h xp h Atx x ππ Ψ ?= The role of E and t in the equation are mathem- atically identical to the role of x and p x . htE ≥?? 3.The implications of the uncertainty principles 1 the uncertainty principle indicate that it is not exact describing the microscopic particle by classical theory. §27.1 The Heisenberg uncertainty principles 4 2the uncertainty principle gives an limitation using classical model. ?Implications of the position-momentum uncertainty principle In classical physics, the position and momentum component can be measured with arbitrary precision in principle, but in quantum mechanics, this is not strictly true, because of the nature of particles. Example 1:Why atoms do not collapse? The minimum momentum of an electron is r h x h pp ≈≈≈ ? ? §27.1 The Heisenberg uncertainty principles The total energy of the electron is r e mr h r e m p PEKEE 0 2 2 2 0 22 4242 πεπε ?=?=+= The value of r minimizes the total energy satisfied 2 2 0 2 0 2 3 2 4 0) 1 ( 4 ) 2 ( 2d d me h r r e rm h r E πε πε = =???= It is same order of magnitude as the fictitious of the Bohr model. §27.1 The Heisenberg uncertainty principles 5 Example 2: You measure the diameter of a shotgun pellet of mass 1.0 g to be 1.00±0.01 mm with a micrometer. What is the minimum uncertainty of its momentum if the magnitude of its momentum is measured simultaneously? Solution: m/skg107 100.1 10626.6 29 5 34 ?×= × × == ? ? ? x h p x ? ? It is exceedingly small. As a result, for such a macroscopic particle, the uncertainty principle has no effect. §27.1 The Heisenberg uncertainty principles 13 pp smkg1.0100100.1 ?? ??=××== vmp Example 3: The accelerating potential difference of TV kinescope is 9kV, the diameter of exit of electron gun is 0.1 mm, is it reasonable to describe the electrons as classical particles? Solution: m/s27.7 101.01011.9 10626.6 331 34 = ××× × == ≥?= ?? ? xm h v hvmxpx x xx ? ? ???? The uncertainty of the speed The speed of the electrons m/s106.5 1011.9 10106.1222 7 31 319 ×≈ × ××× === ? ? m Ve m KE v ? §27.1 The Heisenberg uncertainty principles 6 Example 4: Minimum energy of a particle in a box —zero point energy 222 22 22 )2( )()( ppp pppp ppp av av =?= ??= ?=? 0 22 )()( 2 22 222 ≠≥= ≥= mL h m p E x h pp ? ? For a particle in a box of length L §27.1 The Heisenberg uncertainty principles Solution: The microscopic particle cannot be rest! Example 5: A proton is known to be in the nucleus of an atom. The size of the nucleus is about 1.0×10 -14 m. Find a. p min =?; b. v min =? For nonrelativeistic situation. c. is it reasonable to express its momentum in classical expression? Solution: a. According to the uncertainty principle m/skg106.6 100.1 10626.6 20 14 34 ?×= × × == ? ? ? x h p ? ? b. The minimum speed of the proton m/s100.4 1067.1 106.6 7 27 20 min ×= × × == ? ? m p v §27.1 The Heisenberg uncertainty principles 7 c. The fraction of the speed of light 0.1 /1 1 13.0 100.3 100.4 22 8 7 ≈ ? = = × × = cv c v γ It is reasonable to express its momentum in classical expression. §27.1 The Heisenberg uncertainty principles ?Implications of the energy-time uncertainty principle a. The mass of fundamental particles According to special relativity mcEmcE ?? 2 rest 2 rest =?= According to the energy-time uncertainty principle htmc ≥?? 2 kg1029.8 )888()103( 10626.6 54 28 34 2 ? ? ×= × × == tc h m ? ? The mean lifetime of a free neutron is 888 s, then §27.1 The Heisenberg uncertainty principles 8 The mass of the neutron can be determined quite precisely because it is a relatively long- lived particle. The values of the masses of very short-lived particles will have intrinsic spread because of the Heisenberg uncertainty principle. b. The nature of a vacuum —vacuum is not empty The idea of a vacuum in quantum physics is quite different from classical idea of a vacuum as simply a volume with nothing in it. §27.1 The Heisenberg uncertainty principles 0 2 2 ≥=?= m p PEEKE Classical mechanics 422242222 , cmcpEcmcpE +±=+= Relativity quantum mechanics E + =mc 2 E - =-mc 2 E=0 E + =mc 2 E - =-mc 2 E=0 Dirac sea: positive energy states are empty, negative energy states are filled fully. 42224222 cmcpcmcpEEE +++=?= ?+ ? §27.1 The Heisenberg uncertainty principles 9 The quantum mechanical vacuum is a seething sea of particle-antiparticle pairs, called virtual particles, since their existence is ephemeral. The pairs of virtual particles well up out of nothing, live for very short time, and then disappear(annihilate each other). According to energy-time uncertainty principle htE ≥?? s1004.4 )103)(1011.9(2 10626.6 2 21 2831 34 2 ? ? ? ×= ×× × = == mc h E h t ? ? For a virtual electron-positron pair §27.1 The Heisenberg uncertainty principles c. The verification of experiment: H. B. G. Casimir effect: Lamb shift: Example: Light of wavelength 632.8 nm is incident on an extremely fast shutter that chops the beam into pulses. The shutter stays open for only 1.5×10 -9 s. What is the approximately minimum range of wavelengths ?λ in the light pulses that pass the shutter? §27.1 The Heisenberg uncertainty principles 10 J104.4 105.1 10626.6 25 9 34 ? ? ? ×= × × == =?≥ t h E htEhtE ? ? ???? since λ? λ ? λ ν 2 hc E c hhE =?== therefore m109.8 )1000.3)(10626.6( )108.632)(104.4( 13 834 925 2 ? ? ?? ×≈ ×× ×× = = hc Eλ? λ? §27.1 The Heisenberg uncertainty principles §27.2 Particle-waves and the wavefunction 1. The strange behavior of microscopic particles 11 We will get one pattern on the distant screen if the particles go through a single slit, but a completely different pattern if a the particles go through two or more slits. Thus we cannot think of the particles as going through either on the slit or the other in the double slits arrangement. The idea of a particle as a discrete billiard ball entity must be abandoned in favor of some abstract particle-wave which can diffract and interfere with it self. §27.2 Particle-waves and the wavefunction Display the property of particle, restrain the property of wave. Display the property of wave, restrain the property of particle. §27.2 Particle-waves and the wavefunction 12 §27.2 Particle-waves and the wavefunction 2. The principle of complementarity Microscopic particles propagate as if they were waves and exchange energy as if they were particles—that’s the wave-particle duality.When we measure the arrival of one of them at a detector, we always measure the energy it delivers to a single point, even though that point may be part of a pattern that could only be created by a wave. In experimental event, such entities of wave do one or the other—they cannot simultaneously manifest the properties of wave and particle. Classical particles have precise path of motion. Quantum particle have no path of motion. hpx x ≥??? ),( x px §27.2 Particle-waves and the wavefunction 13 3. Wavefunction How are the particle-waves to be represented in a mathematical context by a wavefunction Ψ(x,t) ? What equation governs the propagation of the particle-waves? §27.2 Particle-waves and the wavefunction ?The wavefunction of an individual particle traveling in one dimension in free space. )(cos 0when )cos(),( 222 kxA t tkxAtx = = ?= Ψ ωΨ The probability of finding a particle in space for a plane monochromatic wave should not depend on the value of x. AAAeeA tkxitkxA Aetx tkxitkxi tkxi *** )]sin()[cos( ),( )()( )( == ?+?= = ??? ? ωω ω ΨΨ ωω Ψ The wavefunction Ψ(x,t) , whatever mathematical form it takes in a particular context, typically is a complex-valued function. §27.2 Particle-waves and the wavefunction 14 ①The standard condition of wavefunctions: Single valued, continuous, and finite. ②Normalized condition: 1d),,,(*),,,( = ∫ ∞ VtzyxΨtzyxΨ 4. The properties of wavefunctions §27.2 Particle-waves and the wavefunction For one-dimension 1d),(*),( = ∫ ∞ xtxΨtxΨ Quantity Ψ*(x,t) Ψ(x,t)dx is usefully defined as the probability of finding the particle in a region of space between x and x+dx at a particular time t. Ψ(x,t)is known as probability amplitude. Since the wavefunction Ψ(x,t) is a complex- valualed function, Ψ(x,t) itself is not a physical quantity that can be observed directly in any experiment; measurements in experiments always yield real numbers. §27.2 Particle-waves and the wavefunction In the quantum domain we lose the ability to predict with certainty what happens to individual particles such as electrons. Notice: 15 §27.3 Operators and the Schr?dinger equation 1. Operators A mathematical operator is a symbolic way of representing a mathematical operation. Notice: Some operators do not commute, but certain operators do commute. x x ? ? and xt ? ? ? ? and 1The momentum operator h p p h k txikAe x tx x tkxi === = ? ? = ? ? ? π λ π ΨΨ ω 22 ),(][),( )( since Ψ ΨΨ Ψ Ψ p x i xi p i x = ? ? ?= ? ? = ? ? h h h then x i ? ? ? h --momentum operator §27.3 Operators and the Schr?dinger equation 16 2 The energy operator ων Ψω Ψ ω h== ?= ? ? = ? ? ? hE iAe tt tkxi ][ )( since then Ψ ΨΨ Ψ Ψ E t i ti E i t = ? ? = ? ? ? ?= ? ? h h h t i ? ? h --- energy operator §27.3 Operators and the Schr?dinger equation Various mathematical operators correspond to physical observable properties, such as momentum and energy. The operators act on the wavefunction, and the resulting eigenvalues are the appropriate values of that physical observable. With the operator formalism, we say that the wavefunction contains all the information about the physical system. This explains why such a premium is placed in quantum mechanics on discovering the form for the wavefunction in a peculiar physical context; the wavefunction Ψ says it all. §27.3 Operators and the Schr?dinger equation 17 §27.3 Operators and the Schr?dinger equation 2. The schr?dinger equation The procedure for finding such an equation was by no means obvious then, nor is it really clear even in hindsight. The best that can be done is to surmise an equation for the particle-waves based on what we know of them and then test the equation and its predictions in situations that can be compared with experiment. The total energy of a particle for nonrelativistic situation ),( 2 2 txV m p E += §27.3 Operators and the Schr?dinger equation Multiply this expression by the wavefunction ),(),()],( 2 [ 2 txEtxtxV m p ΨΨ =+ t iEand x ip ? ? → ? ? ?→ hh By using the operators of p and E t tx itxtxV x i x i m ? ? =+ ? ? ? ? ? ? ),( ),()],())(( 2 1 [ Ψ Ψ hhh t tx itxtxV x tx m ? ? =+ ? ? ? ),( ),(),( ),( 2 2 22 Ψ Ψ Ψ h h 18 §27.3 Operators and the Schr?dinger equation This is called one-dimensional Schr?dinger equation. t tx itxtxV x tx m ? ? =+ ? ? ? ),( ),(),( ),( 2 2 22 Ψ Ψ Ψ h h HtxV xm =+ ? ? ? ),( 2 2 22 h Is called Hamiltonian of the system. t tx itxH ? ? = ),( ),( Ψ Ψ h