(d) A linear perturbation H' = nx is applied to the system. What are the first order energy corrections to the energy eigenvalues E? (e) An anharmonic energy perturbation is applied to the system such that A' = nx*. What (1) is the first order energy correction E for the ground state |0)? NOTE: Only do the ground state!!!
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- Be *(1) the position operator for a particle subjected to a potential of a one-dimensional harmonic oscillator P mox (Ĥ =+ 2m 2 Evaluate [î(t),î(0)] Heisenberg's chart in2) ( Consider an electron trapped in a one-dimensional anharmonic potential. The Hamiltonian for this system is given as: H mw?£? + 2m + B&3 . Regarding the cubic term as being small, apply non- degenerate perturbation theory by letting the perturbation be: A, = ß2³. а) Calculate to first order the ground state energy of this anharmonic oscillator. b) ( | Calculate to second order the ground state energy of this anharmonic oscillator. c) ( Find the lowest order correction to the ground state wave function.40. The first excited state of the harmonic oscillator has a wave function of the form y(x) = Axe-ax². (a) Follow the
- please answer c) only 2. a) A spinless particle, mass m, is confined to a two-dimensional box of length L. The stationary Schrödinger equation is - +a) v(x, y) = Ev(x, y), for 0 < r, y < L. The bound- ary conditions on ý are that it vanishes at the edges of the box. Verify that solutions are given by 2 v(1, y) sin L where n., ny = 1,2..., and find the corresponding energy. Let L and m be such that h'n?/(2mL²) = 1 eV. How many states of the system have energies between 9 eV and 24 eV? b) We now consider a macroscopic box (L of order cm) so that h'n?/(2mL?) ~ 10-20 eV. If we define the wave vector k as ("", ""), show that the density of states g(k), defined such that the number of states with |k| between k and k +dk is given by g(k)dk, is Ak 9(k) = 27 c) Use the expression for g(k) to show that at room temperature the partition function for the translational energy of a particle in a macroscopic 2-dimensional box is Z1 = Aoq, where 2/3 oq = ng = mk„T/2nh?. Hence show that the average…A neutron of mass m of energy E a,V(x) = Vo ) II. Estimate the kinetic energy of the neutron when they reach region III.Consider a system spin-1/2 system, denoted by A, interacting with another system spin-1/2 system, denoted by B, such that the state of the combined system is AB) a++ B|-+). Find (a) the density matrix PA for system A corresponding to this state and (b) obtain the formulas for (()).
- The Hamiltonian for the one dimensional quantum oscillator is 1 p² 1 Ĥ = 1² + ½ k²² = 12 + √ mw² ಠ2m 2m 2 where k = mw². 1) Define the operators ₁₁ and ₁₁ such that Ĥ = ½ħw (p² + ²). Define Ĥ2 as a function of 1 and p₁ such that Ĥ = hwĤ₂. - 2) Let us define the new operators â (1 + i₁) and ↠= ½(î₁ — ip₁). Express ₁ and p₁ as a function of â and â³. Knowing that [^^1,î₁] = i and [1, 1] = -i, calculate âât and â†â. Express Ĥ2 as a function of a and at. 3) Let us define Ñ such that Ĥ₂ = Ñ + ½. Knowing that Ĥ, Ĥ₂ and Ñ have the same eigenstates, what are their corresponding eigenvalues?6I need the answer as soon as possible