you please explain the answer in more details
thank you
- Here Is The Question And My Answers A B And C So Could You Please Explain The Answer In More Details Thank You A The L 1 (68.93 KiB) Viewed 42 times
- Here Is The Question And My Answers A B And C So Could You Please Explain The Answer In More Details Thank You A The L 2 (105.11 KiB) Viewed 42 times
on
Using both particles' spatial coordinates as
as well as t: Ψ(r1, r2, t), satisfying i∂Ψ = HΨ, 2 2 2 2
∂t
where H = −2m1 ∇1 − 2m2 ∇2 + V (r1, r2, t), and d3r1d3r2 |Ψ(r1, r2,
t)|2 = 1.
If V is time independent, we may separate the temporal and spatial
variables,
yielding (Ψ(r1, r2, t) = ψ(r1, r2) exp(−iEt/).
b. (r1, r2) = a(r1)b(r2). If we replace the preceding w.f. with the
following w.f., this
indeterminacy is accurately conveyed. byψ(r1, r2) = ψa(r1)ψb(r2) ±
ψb(r1)ψa(
If we replace the preceding w.f. with the following w.f., this
indeterminacy is
accurately conveyed. byψ(r1, r2) = ψa(r1)ψb(r2) ±
ψb(r1)ψa(
All particles with integer spins are bosons, while all particles
with half-integer spins
are fermions. It follows from the w.fshape. .'s because two
identical fermions cannot
occupy the same state (e.g., a) since the 'antisymmetric'
wavefunction would be
zero. This is the Pauli exclusion principle in action.
c. f = Pf(r1,r2) (r2,r1). Because P2 = 1, the e.v. when P = 1.
This symmetry feature,
like its spin, is fundamental to a particle and cannot be
modified.
(1) identifiable particles; (2) bosons and (3) fermions that are
indistinguishable
particles
(1) discernible particles,
a(x1)b(x2) = a(x1)b(x2):
(x1 x2)2d = x2a + x2b = 2xaxb.
(2,3) indistinguishable particles,
ψ(x1, x2) = ψa(x1)ψb(x2) ± ψb(x1)ψa(x2):
〈(x −x)2〉 =〈(x −x )2〉 ∓2|〈x〉 |2,where 1 2 ± 1 2 d ab
〈x〉ab ≡ dx xψa∗(x)ψb(x).
is ψ0(r1, r2) = ψ100(r1)ψ100(r2) = 83 e−2(r1+r2)/a and
E0 = −109 eV. Πa
Bosons are located closer to one another than identifiable
particles, while fermions are found more apart, according to
physicists. This is due to the lack of interaction between particle
spin and position, resulting in the separation of these
coordinates.
A symmetric spin state (for example, a triplet) must be linked to
an antisymmetric spatial w.f. (This is an antibonding
combination)
2. Consider two identical particles of spin 1/2 that are confined in an 3D isotropic harmonic oscillator potential with the frequency oo, (a) find the ground state energy and the corresponding total wave function of this system when the two particles do not interact. (Hint: It is convenient to construct the total wave functions in term of symmetric or antisymmetric total special wave function and total spin wave functions, i.e., Y,S₁,2, 5₂ ) = - (VG)Xs (5,5), and the (Ys (F₁, F₂) X₁ ( 5₁, 5₂₁) spatial wavefunction for a particle in an isotropic harmonic oscillator potential is given by maj² Ynim (F) = R (r) Ym (0,0)=r¹ fm (r)e Ym (0,0)). (b) Consider now that there exists a 2h Im
weakly attractive spin-dependent potential between the two particles. V(₁,S₁,₂,S₂) = -krir2 - AS$22, where k and 2 are two small positive real numbers. Find the ground state energy of the time-independent perturbed system up to the first-order perturbation. n! 2n+1-a² (Hint: x2+1 e dx = (a>0, n = 1,2,3..). (c) Use the variational method to estimate the 2q²+1 0 unperturbed ground state energy of this system. How does your result compare with that obtained in (a)? (Hint: (1) the Hamiltonian for two non-interacting particle in the isotropic harmonic oscillator potential is Ĥ= Ĥ, + H₂, (2) based on the unperturbed ground state feature discussed in (a), it is convenient to choose a Gaussian-type of radial function as a trial wavefunction with a normalization factor A and an adjustable parameter a, e.g., 1 T trial (7₁1, 5₁,7₂,5₂) = Ae¯ª(²+²) 21 (5,52) and (3) f 1) Jew²x = ²√², [x²e-²x = √² dx=- dx=- 1 4a Va 2 Va 0 -ax² 3 T dx = 8a² -) (25 points) a "