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

In summary, the rotation invariance of the Schrodinger equation can be proven by showing that the equation is invariant under O(3)-rotation if and only if the commutator of the angular momentum operator and the Hamiltonian operator is equal to zero. This can be shown by using the fact that the partial derivatives of the potential, V(x), must satisfy a specific condition for the equation to be rotation invariant.
  • #1
torehan
41
0
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
 
Last edited:
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  • #2
torehan said:
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;

[tex]x_{i} \partial_{k}V(x) = x_{k} \partial_{i}V(x)[/tex]

Proof

[tex]
\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]
[/tex]

Ok, you finish off the proof.

sam
 
  • #3
?

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.
 

1. What is the significance of rotation invariance in the Schrödinger equation?

Rotation invariance in the Schrödinger equation means that the equation maintains its form and solutions remain valid when the coordinate system is rotated. This is important because it allows for the study of physical systems from different perspectives, without changing the underlying physics.

2. How is the rotation invariance of the Schrödinger equation mathematically expressed?

The rotation invariance of the Schrödinger equation is expressed through the commutator relationship [Lx, H] = 0, where Lx is the angular momentum operator and H is the Hamiltonian. This means that the angular momentum operator and the Hamiltonian commute, or that their order in a mathematical expression does not affect the result.

3. What is the physical significance of the commutator [Lx, H] = 0?

The commutator [Lx, H] = 0 implies that the angular momentum and energy of a physical system are conserved quantities, meaning they do not change over time. This is a fundamental property of many physical systems and is a consequence of rotational invariance.

4. How does rotation invariance affect the solutions of the Schrödinger equation?

Rotation invariance allows for the degeneracy of energy levels in the solutions of the Schrödinger equation. This means that multiple states can have the same energy, resulting in a more complex and diverse set of possible solutions for a given physical system.

5. Can the Schrödinger equation be solved without considering rotation invariance?

Yes, the Schrödinger equation can be solved without considering rotation invariance. However, this would only apply to systems that do not exhibit rotational symmetry. For systems with rotational symmetry, ignoring rotation invariance would result in incorrect solutions and a failure to fully understand the system's behavior.

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