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Problem 3: Quantum states in a double delta potential well Consider a particle of mass m in a double delta potential well characterized by the Hamiltonian hat(H)=-(ℏ^(2))/(2m)(d^(2))/(dx^(2))-alpha delta (x+(l)/(2))-alpha delta (x-(l)/(2)) where l is the distance between the wells. The bound states of the isolated left and right wells obey, hat(H)_(L)psi _(L)=E_(L)psi _(L),;,hat(H)_(L)=-(ℏ^(2))/(2m)(d^(2))/(dx^(2))-alpha delta (x+(l)/(2)) and hat(H)_(R)psi _(R)=E_(R)psi _(R),;,hat(H)_(R)=-(ℏ^(2))/(2m)(d^(2))/(dx^(2))-alpha delta (x-(l)/(2)) respectively. The single delta-potential well was solved in class (see Lecture 6). The eigenstates of H_(L) and H_(R) are, psi _(L)(x)=(sqrt(malpha ))/(ℏ)e^(-malpha |x+(l)/(2)(||)/(ℏ^(2))|),;,psi _(R)(x)=(sqrt(malpha ))/(ℏ)e^(-malpha |x-(l)/(2)(||)/(ℏ^(2))|),;,E_(L)=E_(R)=-(malpha ^(2))/(2ℏ^(2)) 1 a. Using the bound states, {psi _(L)(x),psi _(R)(x)}, of the single delta-potential wells as a basis, find the matrix representation of the double delta-potential well characterized by the Hamiltonian given in Eq. (1) tilde(H)=([H_(LL),H_(LR)],[H_(RL),H_(RR)]),tilde(H)_(ij)=(:psi _(i)|(hat(H))|psi _(j):),i=L,R,;,j=L,R Assume that the wells are well separated from each other, so that (:psi _(L)|delta (x-(l)/(2))|psi _(L):) (:psi _(R)|delta (x+(l)/(2))|psi _(R):)0. b. Find the eigenenergies of the double-well potential (i.e., the eigenvalues of tilde(H)).

          Problem 3: Quantum states in a double delta potential well
Consider a particle of mass m in a double delta potential well characterized by the Hamiltonian
hat(H)=-(ℏ^(2))/(2m)(d^(2))/(dx^(2))-alpha delta (x+(l)/(2))-alpha delta (x-(l)/(2))
where l is the distance between the wells. The bound states of the isolated left and right wells obey,
hat(H)_(L)psi _(L)=E_(L)psi _(L),;,hat(H)_(L)=-(ℏ^(2))/(2m)(d^(2))/(dx^(2))-alpha delta (x+(l)/(2))
and
hat(H)_(R)psi _(R)=E_(R)psi _(R),;,hat(H)_(R)=-(ℏ^(2))/(2m)(d^(2))/(dx^(2))-alpha delta (x-(l)/(2))
respectively. The single delta-potential well was solved in class (see Lecture 6). The eigenstates of H_(L) and H_(R) are,
psi _(L)(x)=(sqrt(malpha ))/(ℏ)e^(-malpha |x+(l)/(2)(||)/(ℏ^(2))|),;,psi _(R)(x)=(sqrt(malpha ))/(ℏ)e^(-malpha |x-(l)/(2)(||)/(ℏ^(2))|),;,E_(L)=E_(R)=-(malpha ^(2))/(2ℏ^(2))
1
a. Using the bound states, {psi _(L)(x),psi _(R)(x)}, of the single delta-potential wells as a basis, find the matrix representation of the double delta-potential well characterized by the Hamiltonian given in Eq. (1)
tilde(H)=([H_(LL),H_(LR)],[H_(RL),H_(RR)]),tilde(H)_(ij)=(:psi _(i)|(hat(H))|psi _(j):),i=L,R,;,j=L,R
Assume that the wells are well separated from each other, so that (:psi _(L)|delta (x-(l)/(2))|psi _(L):)~~ (:psi _(R)|delta (x+(l)/(2))|psi _(R):)~~0.
b. Find the eigenenergies of the double-well potential (i.e., the eigenvalues of tilde(H)).

problem 3 quantum states in a double delta potential well consider a particle of mass m in a double delta potential well characterized by the hamiltonian hath 22md2dx2 alpha delta xl2 alpha 75702

Added by Donald V.

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