Rotation Invariant of Sch. Equation and [Lx,H]=0] ?

[torehan]

Rotation Invariant of Sch. Equation and [Lx,H]=0] ??

Here is my first post,

How can I prove that the rotation invariability of the Sch. equations?

How can I prove that [Lx,H]=0

 

[samalkhaiat]

Here is my first post,

How can I prove that the rotation invariability of the Sch. equations?

How can I prove that [Lx,H]=0

Theorem: Schrodinger equation is invariant under O(3)-rotation if and only if;

$$x_{i} \partial_{k}V(x) = x_{k} \partial_{i}V(x)$$

Proof

$$\left[ L_{i} , H \right] = \epsilon_{ijk} \left[x_{j}p_{k} , p^{2} + V(x) \right] = \epsilon_{ijk}x_{j} \left[ p_{k} , V(x) \right]$$

Ok, you finish off the proof.

sam

 

[denian]

?

I would like to address the content and provide my response. The concept of rotation invariance in the Schrödinger equation is a fundamental principle in quantum mechanics. It states that the physical laws and properties of a system should remain unchanged under rotations. In other words, if we rotate the system, the equations that describe it should remain the same.

To prove the rotation invariance of the Schrödinger equation, we can use the transformation properties of the wavefunction. The wavefunction is a mathematical representation of the state of a quantum system and it transforms under rotations according to certain rules. By applying these rules, we can show that the Schrödinger equation remains unchanged under rotations.

Now, let's address the second part of the content, [Lx,H]=0. Here, Lx is the x-component of the angular momentum operator and H is the Hamiltonian operator. This equation represents the commutator between the two operators, which is a mathematical way of measuring how two operators behave with each other.

The fact that [Lx,H]=0 means that the x-component of the angular momentum operator and the Hamiltonian operator commute with each other. In simpler terms, this means that they can be measured simultaneously without affecting each other's values. This is a crucial property in quantum mechanics, as it allows us to determine the state of a system accurately.

To prove this, we can use the definition of the commutator and the properties of the operators involved. By doing so, we can show that the commutator evaluates to zero, thus proving the statement [Lx,H]=0.

In conclusion, both the rotation invariance of the Schrödinger equation and the commutativity of the angular momentum and Hamiltonian operators are essential concepts in quantum mechanics. By understanding and proving these concepts, we can gain a deeper understanding of the behavior of quantum systems and make accurate predictions about their states.