Classical harmonic oscillator Linear force acting on a particle (Hooke s law): F =!kx From Newton s law: F = ma = m d x dt =!kx " d x dt + # x = 0, # = k / m Position and momentum solutions oscillate in time: x(t) = x(0)cos(!t) + p(0) m! sin(!t) p(t) = m dx dt = p(0)cos (!t) " m! x(0)sin(!t)
Classical harmonic oscillator Classical Hamiltonian H = T + V V(x) T = p m F =! "V "x # V = 1 kx = 1 m x x H = p m + 1 m! x
Quantum Hamiltonian: replace x and p variables with operators Define a dimensionless operator Then H = T + V = p m + 1 m! x Position, momentum operators obey the canonical commutation relation: [x, p] = i! a = m!! x + i 1 m!! p a = " # m!! x + i 1 m!! p % & = m!! x ( i 1 m!! p
Commutation relation:!" a,a # =!& m%! x + i 1 ( m%! p ) + *, & m%! x, i 1 ( m%! p ) # - +. "- *. = m% i i [x, x] + [ p, x],!!! [x, p] + 1 [ p, p] m%! = i! (,i!), i! (i!) = 1!" a,a # = 1!" a,a# = %1
Number operator: N = a a = # % = 1 #!! % m!! x "i 1 m!! p & # ( % p m + 1 m! x & ( "1 m!! x + i 1 m!! p & ( Hence we can rewrite the Hamiltonian in terms of the number operator: N = a a = # % m!! x " i 1 m!! p & # ( % H = p m + 1 m! x # =!! a a + 1 & % ( m!! x + i 1 m!! p & (
Number operator: [N, H ] =!" a a, H # =!%! a a, & a a + 1 ) #, ( * + - " = 0 Commutation relations N = ( a a) = a ( a ) = a a = N [N,a] =!" a a,a# = a [ a,a] +!" a,a# a = %a [N,a ] =!" a a,a # = a!" a,a # +!" a,a # a = a [H,a] =!! [ N + 1,a] =!! [ N,a] = "!!a [H,a ] =!! # N + 1,a % & =!! # N,a % & =!!a
Eigenvalues and eigenfunctions The energy eigenfunctions and eigenvalues can be found by analytically solving the TISE. Here we will use operator algebra: H = p m + 1 m! x " =!! N + 1 % # & =!! " a a + 1 % # & Energy eigenvalue equation (TISE): H! n = E n! n The parentheses around ψ are standard (Dirac) notation for states that is independent of x or p representation. More on this notation later. Notice that: Ha! n ( )a! n ( )a! n = ( ah "!#a)! n = E n "!# Ha! n = ( a H +!#a )! n = E n +!#
Eigenvalues and eigenfunctions H! n = E n! n Ha! n = ( E n "!# )a! n Ha! n = ( E n +!# )a! n The state a! n is an energy eigenfunction with eigenvalue ( E n!!" ) The state a! n is an energy eigenfunction with eigenvalue ( E n +!! ) Hence a and a + are called the raising and lowering (ladder) operators since they raise or lower the energy by a definite amount.
Eigenvalues and eigenfunctions H! n = E n! n Consider the lowest eigenvalue of H (ground state energy): H! 0 = E 0! 0 The lowering operator a cannot lower the energy of this eigenstate any further. Hence, a! 0 = 0 (!!a )a " =!! N " = H #!! 0 0 % & ( ) " = E #!! 0 % & 0 ( ) " 0 = 0 Ground state energy: E 0 =!!
Eigenvalues and eigenfunctions H! n = E n! n H! 0 = E 0! 0 E 0 =!! We have seen that the states a! n are energy eigenstates with energy. ( E n +!! ) Thus starting with the lowest energy E 0, the energy eigenvalues are E 0,E 0 +!!,E 0 +!!,... " E n = E 0 + n!! = n + 1 % # &!!
Eigenvalues and eigenfunctions! E n = n + 1 " # % &! A unique feature of the quantum harmonic oscillator is that the energy eigenvalues are equally spaced:
Consider the ground state: The lowering operator a cannot lower the energy of this eigenstate any further. Hence, a! 0 = 0! m"! # 0 +! m" # 0 x = 0 Normalized solution:! 0 (x) = N 0 e " x x c N 0 = m#!, x c =! m#
Now we can calculate the higher energy (excited) states:! 0 (x) = N 0 e " x x c N 0 = m#!, x c =! m# x! 1 (x) = N 1 a! 0 (x) = N 1 " x c % & x c # #x ( )! (x) 0! (x) = N a! n (x) = N n a % & ( )! 0 (x) = N x % & ( ) n! 0 (x) = N n x " x c x c " x c x c # #x ( ) # #x ( ) n! 0 (x)! 0 (x)
Now we can calculate the higher energy (excited) states:! 0 (x) = N 0 e " x x c N 0 = m#!, x c =! m# Normalized solutions:! n (x) = 1 n n! H n y ( )! 0 (x) H n (y): Hermite polynomials y = x x c
First few Hermite polynomials: H 0 (x) = 1, H 1 (x) = x, H (x) = 4x! There are many properties known about Hermite polynomials. See http://mathworld.wolfram.com/hermitepolynomial.html or your favourite mathematics book of special functions for more.
Ground State Expectation values (verify this using the ladder operators, a and a +. See Example.5 in the textbook)!x 0 = x 0 " x =! m#!p 0 = p 0 " p = m#!!x 0!p 0 =! The ground state is a minimum uncertainty state. Recall that such a state must be Gaussian.