Solid-State and molecular physics; allowed energy states

[mjmontgo]

When two objects move under the influence of their mutual force alone, we can treat the relative motion as a one-particle system of mass μ=m₁m₂/(m₁+m₂). An object of mass m₂and charge -e orbits an object of mass m₁ and charge +Ze. By appropriate substitutions into formulas given in the chapter, show that a) the allowed energies are : (Z²μ/m)E₁/n², where E₁ is the hydrogen ground state, and b) the "bohr radius" for this system is (m/Zμ)a₀, where a₀ is the hyrogen bohr radius

Homework Equations

E_n= -me⁴/2(4πε₀)²hbar²(1/n²); n=1,2,3...

L=$\sqrt{l(l+1)}$hbar

L_z=m_Lhbar ; m_L=...-2,-1,0,1,2,...

-hbar²/2m(∇²ψ(r) + U(r)ψ(r)= Eψ(r)

U(r)= -1/(4πε₀)(Ze)(e)/r ---->
E_n= - me⁴Z²/2(4πε₀)²hbar²*(1/n²)


NOTE: hbar= plancs constant divided by 2 pi.

The Attempt at a Solution

Im at a standstill just starting this problem, i know its a long shot asking for help. But if anyone looks at this and has an idea it would be greatly appreciated. Just looking and playing around with substitution i did not get anything close... even some insight or ideas would be great thanks.

 

[mark726]

Thank you for your post. I can help you with this problem. Let's start by looking at the given equation for the allowed energies: En= - me4Z2/2(4πε0)2hbar2*(1/n2). We can see that this equation is similar to the equation for the hydrogen ground state energy, except for the Z2 term. This suggests that Z2 is related to the charge of the orbiting object, -e, and the charge of the central object, +Ze.

Now let's look at the Bohr radius equation: a0=4πε0hbar2/(me2). Again, we can see that this equation is similar to the hydrogen Bohr radius equation, except for the substitution of μ for m. This suggests that μ is related to the masses of the two objects, m1 and m2.

To show that the allowed energies are (Z2μ/m)E1/n2, we can substitute the given values for Z, μ, and m into the hydrogen ground state energy equation. We get:

En= - me4Z2/2(4πε0)2hbar2*(1/n2)

= - (m1m2)4(-e)2/2(4πε0)2hbar2*(1/n2)

= (m1m2)(-e)2/2(4πε0)2hbar2*(1/n2)

= (Z2μ/m)E1/n2

For the "Bohr radius" equation, we can substitute the given values for μ and m into the hydrogen Bohr radius equation. We get:

a0=4πε0hbar2/(me2)

= 4πε0hbar2/((m1m2)2e2)

= (m1m2)2/(4πε0hbar2e2)

= (m1m2)2/((4πε0hbar2)(-e)2)

= (m1m2)2/(4πε0hbar2)(Ze)(e2)

= (m1m2)/(4πε0hbar2)(Ze)

= (m/Zμ)a0

Therefore, the "Bohr radius" for this system is (m/Zμ)a0, where a0 is