Creation and Destruction Operators and Coherent States WKB Method for Ground State Wave Function We first rewrite the ground state harmonic oscillator wave function, m! m!x2 < xj0 >= ( )1=4 exp(− a ) (1) πh¯ 2¯h In the notes on imaginary time path integrals, we obtained this formula from the imag- inary time propagator for the harmonic oscillator. Before introducing creation and de- struction operators, let us explore the exponential factor in the ground state wave func- tions. The method used is called the WKB (for Wentzel, Kramers, and Brillouin) method and involves a quantity closely related to the action used in the path integral discussion. We start by assuming the following form for a bound state wave function, W (x) Ψ = exp(− ): (2) h¯ Note that W has the same dimensions ash ¯ and the action S: Then we have @ W @ Ψ = − x Ψ; x h¯ and " # @ W @2W @2Ψ = ( x )2 − x Ψ x h¯ h¯ Let us use this formula in the Schr¨odingerequation, allowing an arbitrary potential for the moment. From ! h¯2 − @2 + V (x) Ψ = EΨ; 2m x we obtain ! (@ W )2 @2W − x + V (x) +h ¯ x Ψ = EΨ (3) 2m 2m Note that the first two terms on the left side are independent ofh: ¯ The method we are using is semiclassical in which case it makes sense to try to solve the equation order by order inh: ¯ Now consider the eigenvalue E; in particular, suppose we are looking for the ground state of the system. For an oscillator the classical lowest energy is zero, and the actual ground state ish!= ¯ 2: Let us follow this suggestion and assume for other potentials as well that for the ground state E ∼ O(¯h): Then if we make the terms independent of h¯ cancel, we get (@ W )2 x = V (x); 2m or q @xW = 2mV (x) which implies that Z x q W (x) = dx0 2mV (x0) x0 The choice of x0 is not completely specified. However, if we are looking for the ground state of the system, it makes sense to choose x0 at the minimum of V (x): Returning to the oscillator, we obtain Z q Z x x 1 W (x) = dx0 2mV (x0) = dx0m!x0 = m!x2; 0 0 2 which when substituted in Eq.(2) gives the correct form of the oscillator ground state wave function. If we go further and match the O(¯h) terms in Eq.(3), assuming E is proportional toh; ¯ we get @2W h¯ x = E: 2m Using W (x) = m!x; this gives h!¯ E = ; 2 which is the correct answer for the oscillator. This general method can be applied to other smooth potentials, but it is only exact for the oscillator. We will not pause to study the function W (x) further. The relation of W to the action S is basically that one trades the time t for the energy E; by making a so-called Legendre transform. More on this can be found in \Classical Mechanics", by H. Goldstein. There is also much more to the WKB method than we have discussed here. It is more typical to use it for highly excited states of a particle moving in a smooth potential. In that case the particle can be in both the classically allowed region and the classically forbidden region. For the classically allowed region the form W Ψ = exp(i ) h¯ is the appropriate one to use, and the factor of i in the exponent is replacedd by -1 when x moves into the classically forbidden region. For more on WKB and semiclassical quantum mechanics in general see Shankar, sec. 16.2, Gottfried and Yan, sec.2.8 and Chapter VII of \Quantum Mechanics," by Landau and Lifshitz. Creation and Destruction Operators Now let us turn to the creation and destruc- tion operators using our solutions of the Heisenberg equations of motion. Rewriting our previous results, P X(t) = X cos !0t + sin !0t; (4) m!0 and P (t) = P cos !0t + m!0X sin !0t: (5) By taking matrix elements of these equations between eigenstates, we found by matching time dependence on both sides, P < n − 1jX(t)jn >= exp(−i!t) < n − 1j(X + i )jn >; n ≥ 1 m! P < n − 1j(X − i )jn >= 0; all n; m! and P < n + 1jX(t)jn >= exp(i!t) < n + 1j(X − i )jn ≥ 0; m! P < n − 1j(X + i )jn >= 0; all n; m! and further there is a set of states with energiesh! ¯ (n + 1=2); n = 0; 1; 2;:::: It is clear from these results that the operator P (X + i ) m! can only lead to jn − 1 > when acting on jn >; while P (X − i ) m! can only lead to jn+1 > when acting on jn > : Further since all states must have positive energy, we must have P (X + i )j0 >= 0 m! Let us multiply these operators. We have P P P 2 1 (X − i )(X + i ) = X2 + − [PX − XP ] m! m! (m!)2 m! Multiplying by m!2=2 and using [PX − XP ] = −ih;¯ we have 1 P P 1 m!2(X − i )(X + i ) = H − h!¯ 2 m! m! 2 Dividing byh!; ¯ we have H 1 m! P P = + (X − i )(X + i ) h!¯ 2 2¯h m! m! The right hand side is dimensionless. We now define r m! P a ≡ (X + i ); 2¯h m! and r m! P ay ≡ (X − i ): 2¯h m! The commutator of a with ay follows from the X; P commutator. We have m! P P i [a; ay] = [(X + i ); (X − i )] = [P; X] = 1: 2¯h m! m! h¯ The harmonic oscillator Hamiltonian in terms of a and ay is 1 H =h! ¯ (aya + ) 2 Rearranging Eqs.(4) and (5), we have [ ] 1 P P X(t) = (X − i ) exp(i!t) + (X + i ) exp(−i!t) ; 2 m! m! or using the definitions of a and ay; we have s h¯ h i X(t) = a exp(−i!t) + ay exp(i!t) (6) 2m! Writing this equation one more time, s h¯ h i X(t) = a(t) + ay(t) ; 2m! so we have a(t) = a exp(−i!t); ay(t) = ay exp(i!t); where a(t) and ay(t) are the Heisenberg operators corresponding to the Schr¨odingerop- erators a and ay: The corresponding expressions for P (t) follow from these, s dX h!¯ h i P (t) = m = −i a exp(−i!t) − ay exp(i!t) (7) dt 2m Expressions like Eqs.(6) and (7) occur in many more general problems, basically any problem that can be expanded in harmonic oscillators. Examples are photons, phonons, and the so-called \vector mesons" of particle physics, such as the W and Z particles. Let us now find the matrix elements of a and ay: Using 1 E =h! ¯ (n + ); n 2 we have h!¯ < nj(H − )jn >=h!n: ¯ 2 But 1 H − h!¯ =h! ¯ (aya); 2 so we have h!n¯ =h! ¯ < njayajn > Now from our results above, we know that acting on jn > a must lead exclusively to jn−1 > so jn−1 > is the only intermediate state that can be reached when we sandwich the identity between ay and a: Cancelling the factor ofh!; ¯ we have n =< njayjn − 1 >< n − 1jajn >= j < n − 1jajn > j2; which gives n = j < n − 1jajn > j2: Choosing the possible phase factor to be 1, we have p < n − 1jajn >= n =< njayjn − 1 > Rewriting these, we have p q ajn >= njn − 1 >; ayjn >= (n + 1)jn + 1 > : which gives ayajn >= njn > so aya is a number (of excitations) operator. Let us show how to construct the states jn > using ay: Writing out the first few states, we have y a j0 > = pj1 > y a j1 > = p2j2 > ayj2 > = 3j3 > . p . ayjn > = n + 1jn + 1 > Using these equations to express everything in terms of j0 >; we have 1 jn >= p (ay)nj0 > n! We can use this last equation to find the wave functions for all the states. Using r m! P ay = (X − i ); 2¯h m! we have 1 m! P < xjn >= p ( )n=2 < xj(X − i )j0 >; n! 2¯h m! or 1 m! n=2 h¯ < xjn >= p ( ) (x − @x) < xj0 > : n! 2¯h m! This formula will generate the explicit form for < xjn > for any n: Coherent States Coherent states are an important class of states that can be realized by any system which can be represented in terms of a harmonic oscillator, or sums of harmonic oscillators. They are the answer to the question, what is the state of a quantum oscillator when it is behaving as classically as possible? As a practical example,the state of photons in a laser is quantum mechanical, but also behaves classically in many ways. Coherent states are relevant to this situation. The definition of a coherent state jα > is that it is an eigenstate of a; the destruction operator, ajα >= αjα >; with < αjα >= 1: From its definition, a is certainly not a self-adjoint operator, so it if has eigenstates at all, the eigenvalues need not be real, and in fact the value of α is any complex number. Minimum Uncertainty It is easy to show that coherent states exist, but let us first note one of their most characteristic properties, namely they have minimum Heisenberg uncertainty. Let us first express X and P in terms of a and ay: We have y X = x0(a + a ) y P = −ip0(a − a ) where s h¯ x0 = s2m! hm!¯ p = 0 2 Now consider various expected values, < αjXjα >; < αjX2jα >; < αjP jα >; and < αjP 2jα >.