Calculate the energy difference between thems=-12 (“spin up”) and (“spin down”) levels of a hydrogen atom in the 1s state when it is placed in a 1.45-T magnetic field in the negative z-direction. Which level, ms=12or role="math" localid="1663990719445" ms=-12, has the lower energy?

Short Answer

Expert verified

The energy difference between them is 1.68×10-4eV. The levelms=12 has lower energy.

Step by step solution

01

Define magnetic interaction energy for a magnetic dipole and spin magnetic moment

Magnetic interaction energy of a magnetic dipole is potential energy stored in the magnetic dipole when it is placed in an external magnetic field. It is given as,

U=-μ.B

whereis the interaction energy,μ is magnetic moment of the dipole andBis the external magnetic field.

The z-component of the spin magnetic momentμz is related toSzby,

μz=-(2.00232)e2mSz

where is spin angular momentum forms as spin magnetic quantum number.

02

Calculate energy separation between the two levels

The 1s state corresponds to orbital quantum number l=0 so, the magnetic quantum number is ml=0.

There is no orbital dipole moment for this state and the interaction with external magnetic field is due to the intrinsic spin of electrons.

The z-component of spin magnetic moment is given as,

μz=-2.00232e2mmsħμz=-(11.589×10-5eV/T)×ms

The angle between z-component of spin magnetic moment and z-component of magnetic field is180° as both are oppositely oriented.

This gives the magnetic interaction energy as,

U=-μzBz×cos180°U=-μzBz×-1U=-11.589×10-5eV/T×msB

From above equation it is clear that interaction energy for spin upms=12 is lower than that of spin down ms=-12.

The spin magnetic quantum numberms takes value either-12 or +12.

So, the interaction energy separation between two levels corresponding to each value ofms is,

ΔE=U(1/2-U-1/2ΔE=2.00232×5.788×10-5eV/T12--12×1.45TΔE=1.68×10-4eV

Therefore, the energy difference between them is 1.68×10-4eV. The levelms=12 has lower energy.

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