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Chapter 8: Q34E (page 340)

Show that the symmetric and anti symmetric combinations of (8鈭�19)and(8鈭�20)are solutions of the two. Particle Schr枚dinger equation(8鈭�13)of the same energy as蠄n(x1)蠄m(x2), the unsymmetrized product(8鈭�17).

Short Answer

Expert verified

It is proved that symmetric and asymmetric combination are the solution of two particles Schrodinger equation.

Step by step solution

01

Given information

The symmetric wave function is蠄s(x1,x2)=蠄n(x1)蠄n(x2)+蠄n(x1)蠄n(x2) .

The asymmetric wave function is蠄A(x1,x2)=蠄n(x1)蠄m(x2)鈭�蠄m(x1)蠄n(x2) .

02

Concept of complex number:

The expression for combination of symmetric and asymmetric wave function is given by 蠄SA(x1,x2)=蠄n(x1)蠄n(x2)卤蠄n(x1)蠄n(x2)

03

Evaluate symmetric and asymmetric wave function

The expression for combination of symmetric and asymmetric wave function is calculated as,

Apply Schrodinger wave equation

By Schrodinger wave equation in one dimension,

Considering equation (1) and (2)

鈭�h22m鈭�2鈭�x12+鈭�2鈭�x22蠄SA(x1,x2)+[U(x1)+U(x2)]蠄SA(x1,x2)=贰蠄蠄SA(x1,x2)

Thus, the symmetric and asymmetric combination obeys the Schrodinger equation with two particle solution.

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Most popular questions from this chapter

Exercise 44 gives an antisymmetric multiparticle state for two particles in a box with opposite spins. Another antisymmetric state with spins opposite and the same quantum numbers is 蠄n(x1)鈫�n2(x2)鈫�鈭�蠄nn(x1)鈫�蠄n(x2)鈫�

Refer to these states as 1 and 11. We have tended to characterize exchange symmetry as to whether the state's sign changes when we swap particle labels. but we could achieve the same result by instead swapping the particles' stares, specifically theandin equation (8-22). In this exercise. we look at swapping only parts of the state-spatial or spin.

(a) What is the exchange symmetric-symmetric (unchanged). antisymmetric (switching sign). or neither-of multiparticle states 1 and Itwith respect to swapping spatial states alone?

(b) Answer the same question. but with respect to swapping spin states/arrows alone.

(c) Show that the algebraic sum of states I and II may be written(蠄n(x1)蠄n'(x2)鈭�蠄n'(x1)蠄n(x2))(鈫�鈫�+鈫�鈫�)

Where the left arrow in any couple represents the spin of particle 1 and the right arrow that of particle?

(d) Answer the same questions as in parts (a) and (b), but for this algebraic sum.

(e) ls the sum of states I and 11 still antisymmetric if we swap the particles' total-spatial plus spin-states?


(f) if the two particles repel each other, would any of the three multiparticle states-l. II. and the sum-be preferred?

Explain.

Here we consider adding two electrons to two "atoms," represented as finite wells. and investigate when the exclusion principle must be taken into account. In the accompanying figure, diagram (a) shows the four lowest-energy wave functions for a double finite well that represents atoms close together. To yield the lowest energy. the first electron added to this system must have wave function Aand is shared equally between the atoms. The second would al so have function Aand be equally shared. but it would have to be opposite spin. A third would have function B. Now consider atoms far a part diagram(b) shows, the bumps do not extend much beyond the atoms - they don't overlap-and functions Aand Bapproach equal energy, as do functions Cand D. Wave functionsAandBin diagram (b) describe essentially identical shapes in the right well. while being opposite in the left well. Because they are of equal energy. sums or differences ofandare now a valid alternative. An electron in a sum or difference would have the same energy as in either alone, so it would be just as "happy" inrole="math" localid="1659956864834" A,B,A+B, orA- B. Argue that in this spread-out situation, electrons can be put in one atom without violating the exclusion principle. no matter what states electrons occupy in the other atom.

A lithium atom has three electrons. These occupy individual particle states corresponding to the sets of four quantum numbers given by .

(n,l,ml,mj)=(1,0,0,+12),(1,0,0,-12)and(2,0,0,+12)

Using 蠄1,0,0(rj)鈫�,蠄1,0,0(rj)鈫�,and蠄2,0,0(rj)鈫� to represent the individual-particle states when occupied by particlej . Apply the Slater determinant discussed in Exercise 42 to find an expression for an antisymmetric multiparticle state. Your answer should be sums of terms like .

蠄1,0,0(r1)鈫�,蠄1,0,0(r2)鈫�,and蠄2,0,0(r3)鈫�

What is the angle between Land Sin a (a) 2p3/2and(b) 2p1/2 state of hydrogen?

What is the minimum possible energy for five (non-interacting) spin -12particles of massmin a one dimensional box of length L ? What if the particles were spin-1? What if the particles were spin -32?
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