Further generalization to 3D case 1 and the probability for the particle to have its momentum between(p Pp)and (p+ dpz:Py +dpy:p:+dp=)is P(p)dp="(p.t)(p.t)dpzdpydp: This can be visualized in momentum space as the probability of p vector to have its endpoint in volume element dp. Note: .Once the wave function (r,t)is known,(p,t)is uniquely defined though Fourier transformation.Thus,the probability distribution of p is known. .(r,t)and (p,t)have the same function in representing a state in this sense,i.e., there is no priority of (r,t)over (p,t),(r,t)is called coordinate representation of the state,while (p,t)is called the momentum representation of the same state. .Just as in coordinate representation,(p,t)represents a probability wave and the numerical value ofp,t)is the probability amplitude at momentum p D.Expectation value of dynamical quantity In ID case we say that the operator r represents position,and the operator ih represents momentum,in quantum mechanics;to calculate expectation values,we sandwich the appropriate operator betweenand and integrate =产dve.ae0 -we(品)e That's cute,but what about other dynamical variables?The fact is,all such quantities can be written in terms of position and momentum.Kinetic energy,for example,is 26
Further generalization to 3D case ψ(r, t) = 1 (2π~) 3/2 ZZZ ϕ(p, t)e i ~ p·r d 3p and the probability for the particle to have its momentum between (px, py, pz) and (px + dpx, py + dpy, pz + dpz) is P(p)d 3p = ϕ ∗ (p, t)ϕ(p, t)dpxdpydpz This can be visualized in momentum space as the probability of p vector to have its endpoint in volume element d 3p. Note: • Once the wave function ψ(r, t) is known, ϕ(p, t) is uniquely defined though Fourier transformation. Thus, the probability distribution of p is known. • ψ(r, t) and ϕ(p, t) have the same function in representing a state in this sense, i.e., there is no priority of ψ(r, t) over ϕ(p, t), ψ(r, t) is called coordinate representation of the state, while ϕ(p, t) is called the momentum representation of the same state. • Just as in coordinate representation, ϕ(p, t) represents a probability wave and the numerical value of ϕ(p, t) is the probability amplitude at momentum p. D. Expectation value of dynamical quantity In 1D case we say that the operator x represents position, and the operator −i~ ∂ ∂x represents momentum, in quantum mechanics; to calculate expectation values, we sandwich the appropriate operator between ψ ∗ and ψ, and integrate hxi = Z +∞ −∞ dxψ∗ (x, t) (x) ψ (x, t) hpxi = Z +∞ −∞ dxψ∗ (x, t) µ −i~ ∂ ∂x¶ ψ (x, t) That’s cute, but what about other dynamical variables? The fact is, all such quantities can be written in terms of position and momentum. Kinetic energy, for example, is T = 1 2 mv2 = p 2 2m 26
and the angular momentum is L=rxp (the latter,of course,does not occur for motion in one dimension).To calculate the expec- tation value of any such quantity,Q(,p),we simply replace every p by-ih,insert the resulting operator between andand integrate (3) Extension to 3D case is straightforward. There are two cornerstones of Quantum Mechanics:The state of motion is described by wave functions while dynamical quantities are represented as operators.They cooperate each other into a complete theory.Thus all physical quantities must be operatorized.The consequences of operatorization of physical quantities have profound significance in quantum theory.We also notice the importance of cartesian coordinate in establishing operator system. V.SCHRODINGER EQUATION A.The quest for a basic equation of quantum mechanics Premise:The state of system is described by a wave function(r,t) Quest:Basic equation for the development of(r,t) What is the rule that (r,t)must obey and what is the rule that governs the development and evolution of the wave function? Answer:Ingenious conjecture Our way:Try to approach the result through a simple example-Free particle,whose wave function is already known (r,t)=et(pr-E) We calculate the first derivatives of the wave function with respect to its variables.The 27
and the angular momentum is L = r × p (the latter, of course, does not occur for motion in one dimension). To calculate the expectation value of any such quantity, Q (x, p), we simply replace every p by −i~ ∂ ∂x , insert the resulting operator between ψ ∗ and ψ, and integrate hQ (x, p)i = Z +∞ −∞ dxψ∗ (x, t) Q µ x, −i~ ∂ ∂x¶ ψ (x, t) (3) Extension to 3D case is straightforward. There are two cornerstones of Quantum Mechanics: The state of motion is described by wave functions while dynamical quantities are represented as operators. They cooperate each other into a complete theory. Thus all physical quantities must be operatorized. The consequences of operatorization of physical quantities have profound significance in quantum theory. We also notice the importance of cartesian coordinate in establishing operator system. V. SCHRODINGER EQUATION ¨ A. The quest for a basic equation of quantum mechanics Premise: The state of system is described by a wave function ψ(r, t) Quest: Basic equation for the development of ψ(r, t) What is the rule that ψ(r, t) must obey and what is the rule that governs the development and evolution of the wave function? Answer: Ingenious conjecture Our way: Try to approach the result through a simple example - Free particle, whose wave function is already known ψ(r, t) = e i ~ (p·r−Et) We calculate the first derivatives of the wave function with respect to its variables. The 27
equation should be of the form of a partial differential equation →h E =京候++=常 1 We have for free particle h品=B0= For a particle in potential V(r)we replace E in the above equation with E=P+V四 and get a generalized equation h2n9-(+v)r (4) t Usually it is written in a more compact form h0=c,) at with the Hamiltonian operator of the system(the operator of energy of the system)defined as 月=2p2+,) 1 =2(房+房+)+V红，，) 1 Equation (4)is called the Schrodinger Equation,one of the basic axioms of quantum me- chanics. We noted that: .The Schrodinger equation (4)is not deduced,but conjectured.Its correctness is qual- ified by the accordance of results deduced from it with experimental facts. 28
equation should be of the form of a partial differential equation ∂ψ ∂t = − i ~ Eψ ⇒ i~ ∂ψ ∂t = Eψ ∂ψ ∂x = i ~ px, ∂ 2ψ ∂x2 = − 1 ~ 2 p 2 xψ ∂ψ ∂y = i ~ py, ∂ 2ψ ∂y2 = − 1 ~ 2 p 2 yψ ∂ψ ∂z = i ~ pz, ∂ 2ψ ∂z2 = − 1 ~ 2 p 2 zψ ∇2ψ = − 1 ~ 2 (p 2 x + p 2 y + p 2 z )ψ = − 2m ~ 2 Eψ We have for free particle i~ ∂ ∂tψ = Eψ = − ~ 2 2m ∇2ψ For a particle in potential V (r) we replace E in the above equation with E = 1 2m p 2 + V (r) and get a generalized equation i~ ∂ψ(r, t) ∂t = µ − ~ 2 2m ∇2 + V (r) ¶ ψ(r, t) (4) Usually it is written in a more compact form i~ ∂ψ(r, t) ∂t = Hψˆ (r, t) with the Hamiltonian operator of the system (the operator of energy of the system) defined as Hˆ = 1 2m ˆp 2 + V (x, y, z) = 1 2m ¡ pˆ 2 x + ˆp 2 y + ˆp 2 z ¢ + V (x, y, z) Equation (4) is called the Schr¨odinger Equation, one of the basic axioms of quantum mechanics. We noted that: • The Schr¨odinger equation (4) is not deduced, but conjectured. Its correctness is qualified by the accordance of results deduced from it with experimental facts. 28
Notice the special role played by Cartesian coordinate system in obtaining the Hamil- tonian. The discussion can be generalized directly to many-body system h元)=，…） a=∑+ve,…)+ 7 where U(r)characterize the interaction between particles. B.Probability current and probability conservation The physical meaning of p(r,t)=(r)(rt)is the probability density.From the Schrodinger equation (4)and its complex conjugate -inDv(r.t) v) we can calculate the time derivative of the probability density h层erc加c,》=三wrpt-vo =-2.(07-vw 2 An integration over a volume V gives h层八e0=-然-eT-产 which can be transferred into an integration over the surfaces enclosed by V and 品瓜er+f（热)ew-西=0 Define now the current density of probability as j=(）- 29
• Notice the special role played by Cartesian coordinate system in obtaining the Hamiltonian. • The discussion can be generalized directly to many-body system i~ ∂ ∂tψ (r1, r2, r3, · · ·) = Hψˆ (r1, r2, r3, · · ·) Hˆ = X i 1 2mi ˆp 2 i + V (r1, r2, r3, · · ·) +X ij U(rij ) where U(rij ) characterize the interaction between particles. B. Probability current and probability conservation The physical meaning of ρ(r, t) = ψ ∗ (r, t)ψ(r, t) is the probability density. From the Schr¨odinger equation (4) and its complex conjugate i~ ∂ψ(r, t) ∂t = µ − ~ 2 2m ∇2 + V (r) ¶ ψ(r, t) −i~ ∂ψ∗ (r, t) ∂t = µ − ~ 2 2m ∇2 + V (r) ¶ ψ ∗ (r, t) we can calculate the time derivative of the probability density i~ ∂ ∂t (ψ ∗ (r, t)ψ(r, t)) = − ~ 2 2m (ψ ∗∇2ψ − ψ∇2ψ ∗ ) = − ~ 2 2m ∇ · (ψ ∗∇ψ − ψ∇ψ ∗ ) An integration over a volume V gives i~ d dt ZZZ V ρ(r, t)d 3 r = − ~ 2 2m ZZZ V ∇ · (ψ ∗∇ψ − ψ∇ψ ∗ )d 3 r which can be transferred into an integration over the surface s enclosed by V i~ d dt ZZZ V ρ(r, t)d 3 r = − ~ 2 2m I s (ψ ∗∇ψ − ψ∇ψ ∗ ) · ds and d dt ZZZ V ρ(r, t)d 3 r + I s µ − i~ 2m ¶ (ψ ∗∇ψ − ψ∇ψ ∗ ) · ds = 0 Define now the current density of probability as j = µ − i~ 2m ¶ (ψ ∗∇ψ − ψ∇ψ ∗ ) 29
FIG.12:Current density of probability we thus arrive at the continuation equation of probability density 益川er+s=0 and the differential and integral equation of probability conservation 0+vj=0 at 品川er=-打s This equation is telling us that the rate of change of density (the amount of a substance at a point)is equal to (-ve of)the gradient (grad)of the current density.If we write this in one dimension it's perhaps a little easier to see why this relationship is true: Essentially,at a point is the difference between the amount of"stuff flowing into that point and the amount flowing ou Since probability is conserved,if there is any difference. this must be the rate of change of the amount of stuff at that point i.e.Op/0t. 30
FIG. 12: Current density of probability we thus arrive at the continuation equation of probability density d dt ZZZ V ρ(r, t)d 3 r + I j · ds = 0 and the differential and integral equation of probability conservation ∂ρ ∂t + ∇ · j = 0 d dt ZZZ V ρ(r, t)d 3 r = − I j · ds This equation is telling us that the rate of change of density (the amount of a substance at a point) is equal to (-ve of ) the gradient (grad) of the current density. If we write this in one dimension it’s perhaps a little easier to see why this relationship is true: ∂ρ ∂t = − ∂j ∂x Essentially, ∂j/∂x at a point is the difference between the amount of “stuff” flowing into that point and the amount flowing out. Since probability is conserved, if there is any difference, this must be the rate of change of the amount of stuff at that point i.e. ∂ρ/∂t. 30