A system that has spin momentum √(6) ħis rotated through an angle ϕaround the z-axis. Write down

A system that has spin momentum √(6) ħis rotated through an...

Key Concepts

Rotation Operator in Quantum Mechanics

This concept refers to the unitary operator that rotates quantum states. In quantum mechanics, rotations are implemented using operators of the form exp(?i?J/?), where J is the angular momentum operator along the axis of rotation and ? is the rotation angle. The rotation operator preserves probabilities and transforms the state vector in Hilbert space while altering its phase according to the eigenvalue of the angular momentum component along the rotation axis.

Spin Angular Momentum Eigenstates

Spin angular momentum eigenstates are the basis states that have definite values of a spin component, usually S_z. For a system with spin quantum number S, the eigenvalues m range from ?S to S in integer steps. These states form a complete, orthonormal basis for the spin space and are used to define the matrix representation of operators acting on spin systems.

Matrix Representation of Rotation Operators

When an operator like a rotation is expressed in the eigenbasis of the spin component (usually S_z), it takes a matrix form where the action on each eigenstate is simply to multiply by a phase factor. This results in a diagonal matrix for rotations about the z-axis, with diagonal entries given by exp(?im?) for each eigenvalue m. Such representations simplify the computation of rotated state amplitudes in quantum mechanics.

Wigner D-Matrix Elements

The Wigner D-matrix elements describe the rotation of angular momentum eigenstates in a compact form. For rotations about the z-axis, these elements reduce to simple exponentials because the operator commutes with S_z. The diagonal structure of the matrix reflecting these elements is especially useful in problems involving spin systems, as it directly relates the rotation angle to a phase factor for each eigenstate.

Transcript

00:01 In this problem, we are going to calculate the angular momentum of the disk and prod system.

00:07 That is hz.

00:11 Let's write the equation for this angular momentum.

00:13 X is equals to minus iz x x omega x minus i xy xy omega y plus i z z z omega z we call it equation number one now let's calculate these moment of inertials and angular velocity so by applying this metric we can white i x y is equals to i y z is equals to i z x is equals to zero similarly we can white i y is equals to i x and this is equals to one divided by four into seven kg this is the mass of the disk into 0 .1 meter whole square plus 7 kg into 0 .2 meter whole square this is the moment of inertia of the disk plus the moment of inertia of the rod which is equals to 1 divided by 3 into 1 .5 kg into 0 .2 meter whole square so from here we can white iy is equals to i x so this is equals to 0 .317 5 kg meter square similarly we can write i z as i z is equals to 1 divided by 2 into 7 kg into 0 .1 meter whole square so this is equals to 0 .0 35 kg meter square.

02:17 Now let's calculate the direction of cosines as for z prime axis.

02:23 We can write uz is equals to cosine of 26 .56 degrees so this is equals to 0 .89944.

02:37 We can write ui as ui is equals to 100 and 16 .57 degree so we can write ui is equals to minus 0 .4472 similarly we can write ux as ux is equals to cosine of 90 degrees so this is equals to 0 let's write the equation for i z prime as i z prime is equal to i x x is equal to i x and ux square plus i y uy uy square plus i z uz square minus 2 i x y x y x u x u x u x u x u x u x u x u x let's write the values into this equation we can write this one as i u z x prime is equal to 0 as this ux is equals to 0 plus here we can write 0 .3175 kgmeter square into minus 0 .4472 whole square plus here we can write 0 .035 kjmeter square into 0 .8944.

04:21 Whole square minus 0 minus 0 minus 0...

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