Chapter2_all - PHY4604 R D Field Theory of Stationary...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
PHY4604 R. D. Field Department of Physics Chapter2_1.doc University of Florida Theory of Stationary States (1) Time Dependent Equation: Look for solutions of the equation (A) t t x i t x H op Ψ = Ψ ) , ( ) , ( h or 0 ) , ( ) ( = Ψ t x t i H op h Stationary States: The stationary states are solutions of the form ) ( ) ( ) , ( t x t x Φ = Ψ ψ with ) ( ) ( x E x H op ψ ψ = and ) ( ) ( t E t t i Φ = Φ h which implies that h / ) ( iEt e t = Φ . Eigenvalue Equation: The eigenstates of the system are determined from the eigenvalue equation ) ( ) ( x E x H n n n op ψ ψ = , with n = 1,2, 3, ... Orthonormal Set: The eigenstates form an orthornormal set of wavefunctions such that ij j i j i j i dx x x dx t x t x δ ψ ψ = = Ψ Ψ >≡ Ψ Ψ < +∞ +∞ ) ( ) ( ) , ( ) , ( | . Superposition Principle: The Hamiltonian operator is a “linear operator” which means that ) , ( ) , ( ) , ( 2 2 1 1 t x c t x c t x Ψ + Ψ = Ψ , is also a solution of (A) , where c 1 and c 2 are arbitrary complex numbers. Proof: 0 ) , ( ) ( ) , ( ) ( )) , ( ) , ( )( ( ) , ( ) ( 2 2 1 1 2 2 1 1 = Ψ + Ψ = Ψ + Ψ = Ψ t x t ih H c t x t ih H c t x c t x c t ih H t x t ih H op op op op Most General Solution: The most general solution of (A) is = = = Ψ = Ψ 1 / 1 ) ( ) , ( ) , ( n t iE n n n n n n e x c t x c t x h ψ . Normalization: The arbitrary complex constants must satisy 1 1 = = n n n c c . Proof: ∑ ∑ = = = = = > < + > < >= Ψ Ψ =< 1 1 / ) ( 1 1 * | | | 1 n n n m t E E i n m n n m n m n n n n n c c e c c c c m n h ψ ψ ψ ψ
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
PHY4604 R. D. Field Department of Physics Chapter2_2.doc University of Florida Theory of Stationary States (2) Probability Density: The probability density, in general, depends on time and is given by ∑ ∑ = = = + = Ψ Ψ = 1 1 1 ) , ( ) , ( ) , ( m t i m n n n m n m n n n n n mn e c c c c t x t x t x ω ψ ψ ψ ψ ρ where h / ) ( n m mn E E = ω and called the “transition” frequencies. Average Energy: The average energy of the arbitrary state = = = Ψ = Ψ 1 / 1 ) ( ) , ( ) , ( n t iE n n n n n n e x c t x c t x h ψ . is = = = >= < 1 2 1 | | n n n n n n n E c E c c E and P n = |c n | 2 is the probability that in a single measurement of the energy of the arbitrary state Ψ one would find E n . Proof: ∑ ∑ = = = = = > < + > < >= Ψ Ψ >=< < 1 * 1 / ) ( 1 * 1 * | | | | n n n n m t E E i j m n n i n n m n n n n n n op E c c e E c c E c c H E m n h ψ ψ ψ ψ Overlap Functions: The complex constants are the overlap of the eigenstate Ψ n with the arbitrary state Ψ since > Ψ Ψ =< | n n c and > Ψ Ψ =< n n c | . P n = |c n | 2 is the probability that in a single measuremen of an arbitrary state Ψ would find it in the eigenstate Ψ n . Time Dependence: Suppose we know Ψ (x,t) at t = 0 . Then we know Ψ (x,t) at all later times! Proof: = = Ψ = = Ψ 1 0 ) ( ) ( ) 0 , ( n n n x c x t x ψ with > Ψ =< 0 | n n c ψ = = Ψ 1 / ) ( ) , ( n t iE n n n e x c t x h ψ
Image of page 2
PHY4604 R. D. Field Department of Physics Chapter2_3.doc University of Florida The Infinite Square Well (1) Particle in a One-Dimensional Box: Consider the solution of ) ( ) ( ) ( ) ( 2 2 2 2 x E x x V dx x d m ψ ψ ψ = + h , where h / ) ( ) , ( iEt e x t x = Ψ ψ , for the case V(x) = if x 0 and V(x) =
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern