Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Quantum Mechanics Lecture 4 Dr. Mauro Ferreira E-mail: [email protected] Room 2.49, Lloyd Institute Postulates of Quantum Mechanics P3: The time evolution of the state (wave) functions follows the Schroedinger equation: ∂ i! Ψ("r, t) = ĤΨ("r, t) ∂t where H is the Hamiltonian operator and corresponds to the total energy of the system P̂ 2 Ĥ = + V̂ (!r, t) 2m !2 2 =− ∇ + V̂ (!r, t) 2m The solution of the fully deterministic differential equation above provides ψ(r,t) for all values of r and t. Solution of partial differential equations (PDE) by the method of separation of variables ∂2Φ ∂2Φ + =0 2 2 ∂x ∂y ( Laplace equation ) Φ(x, y) = X(x)Y (y) ( attempted solution ) 1 d2 X 1 d2 Y + =0 2 2 X dx Y dy 1 d2 X 1 d2 Y =− = const 2 2 X dx Y dy The only way these two terms can satisfy the equation is if each one of them is a constant 1 d2 X = const 2 X dx 1 d2 Y = −const 2 Y dy The two-variable P.D.E. was transformed into two O.D.E. ∂ !2 2 i! Ψ("r, t) = − ∇ Ψ("r, t) + V ("r, t)Ψ("r, t) ∂t 2m How to solve the Schroedinger Equation: ( V (!r, t) = V (!r) ) ∂ !2 2 i! Ψ("r, t) = − ∇ Ψ("r, t) + V ("r)Ψ("r, t) ∂t 2m Typically solved by the method of separation of variables Ψ(!r, t) = ψ(!r) φ(t) (attempted solution) 2 simple ODEs { ! r) ψ(!r) = E ψ(!r) H(! dφ(t) i! = E φ(t) dt Eigenvalue equation (Time-independent Schroedinger Eq.) Note that the separation constant E is the energy { ! r) ψ(!r) = E ψ(!r) H(! − !i Et dφ(t) i! = E φ(t) dt φ(t) = e Time-dependence of the wave function (conservative potential) − !i Et Ψ(!r, t) = ψ(!r) e Suppose the system is in an eigenstate Ei of the Hamiltonian. ! r) ψE (!r) = Ei ψE (!r) In other words, H(! i i − !i Ei t Ψ(!r, t) = ψEi (!r) e ⇒ |Ψ(!r, t)|2 = |ψ(!r)|2 • Probability distribution is independent of time. • Eigenstate of the Hamiltonian is called a stationary state Being a linear differential equation, the Schroedinger Eq. must obey the superposition principle, i.e., a linear combination of solutions is also a solution. Consider a state initially described by Ψ(!r, t = 0) = am ψEm (!r) + an ψEn (!r) A short time later, the state will have evolved into − !i Em t Ψ(!r, t) = am ψEm (!r) e − !i En t + an ψEn (!r) e Regarding the probability density function |Ψ(!r, t)|2 = |am ψEm (!r)|2 + |an ψEn (!r)|2 + a∗m − !i (En −Em )t ∗ an ψEm ("r) ψEn ("r) e + am a∗n i ∗ ! ψEm ("r) ψEn ("r) e (En −Em )t It is no longer stationary Another way of seeing the stationary character of the solutions is by calculating the uncertainty of the energy E, which is a solution of the eigenvalue equation Ĥ ψ(x) = E ψ(x) ! ∞ ! ∞ !H" = dx ψ ∗ (x) Ĥ ψ(x) = E dx ψ ∗ (x) ψ(x) = E −∞ −∞ Bearing in mind that Ĥ 2 ψ(x) = Ĥ ( Ĥ ψ(x) ) = Ĥ ( E ψ(x) ) = E 2 ψ(x) !H 2 " = ! ∞ dx ψ ∗ (x) Ĥ 2 ψ(x) = E 2 −∞ ∆H = ! !H 2 " − !H"2 = 0 If the uncertainty on the energy vanishes, the characteristic time interval diverges ( ∆E ∆t ≥ !/2 )