Physical interpretation of the rotation of a ket

• bjnartowt
Therefore, in summary, rotating a quantum -ket refers to applying a rotation operator to the state vector, resulting in a new state vector that represents the state of the system after the rotation. This is similar to rotating a coordinate system, except in this case, it is the state of the system that is being rotated.
bjnartowt
physical interpretation of the "rotation" of a ket

Homework Statement

No problem statement. I'm just having trouble imagining what it means to "rotate" a quantum -ket, especially since not all -kets are eigenstates of position. I know what the rotation operator is. I also can picture rotating a coordinate system in my head--that's easy of course. But what does it mean when a -ket is rotated?

Homework Equations

see attached .pdf. they are notes I've taken from shankar p. 306-306.

The Attempt at a Solution

see attached .pdf. they are notes I've taken from shankar p. 306-306.

Attachments

• 306 - rotations in two dimensions.pdf
14.6 KB · Views: 259

Take the spin of an electron, for example. Suppose it's in the state
$$\vert \alpha \rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1 \end{pmatrix}$$
which is the +-eigenstate of Sx. If you rotate it by 90 degrees about the z-axis, you'd expect it to be in the +-eigenstate of Sy. The rotation operator about the z-axis for spin-1/2 is
$$R(\phi)=\exp\left(-\frac{i S_z \phi}{\hbar}\right) = \begin{pmatrix}e^{-i\phi/2} & 0 \\ 0 & e^{i\phi/2}\end{pmatrix}$$
For $\phi=\pi/2$, you get
$$\vert \alpha_R \rangle = R(\pi/2)\vert \alpha \rangle = \frac{1}{\sqrt{2}} \begin{pmatrix}e^{-i\pi/4} \\ e^{i\pi/4}\end{pmatrix} = \frac{e^{-i\pi/4}}{\sqrt{2}} \begin{pmatrix} 1 \\ i \end{pmatrix}$$
which is indeed the +-eigenstate of Sy.

What is the physical interpretation of the rotation of a ket?

The physical interpretation of the rotation of a ket is the change in its orientation or direction in physical space. It is a mathematical representation of a physical rotation.

How is the rotation of a ket represented mathematically?

The rotation of a ket is represented by a unitary operator, which is a mathematical operation that preserves the length and inner product of the ket. This operator is commonly denoted by a capital letter "U" and is applied to the original ket to produce the rotated ket.

What is the significance of the unitary operator in the rotation of a ket?

The unitary operator in the rotation of a ket ensures that the physical properties of the ket, such as its length and inner product, are preserved. This is important because it allows for a consistent and accurate representation of the physical rotation.

Can the rotation of a ket be physically observed?

No, the rotation of a ket is a mathematical concept that represents a physical rotation in space. It cannot be physically observed, but its effects can be observed through measurements and experiments.

Are there any real-life applications of understanding the physical interpretation of the rotation of a ket?

Yes, understanding the rotation of a ket is crucial in fields such as physics, chemistry, and engineering. It is used to accurately describe and predict the behavior of physical systems, such as the rotation of molecules or the movement of particles in quantum mechanics.

Replies
3
Views
1K
Replies
2
Views
3K
Replies
16
Views
2K
Replies
5
Views
10K
Replies
4
Views
3K
Replies
4
Views
6K
Replies
4
Views
3K
Replies
6
Views
956
Replies
32
Views
2K
Replies
2
Views
2K