Tensor product in Cartesian coordinates

In summary, when discussing a 2d quantum system, the operators acting on the Hilbert space are built using the tensor product of the base kets of two independent degrees of freedom. This means that expressions such as ##V'=\alpha xy## can be written as ##V'=\alpha x \otimes y##, as is shown in Cohen-Tannoudji's quantum mechanics textbook.
  • #1
LagrangeEuler
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I am confused. Why sometimes perturbation ##V'=\alpha xy## we can write as ##V'=\alpha x \otimes y##. I am confused because ##\otimes## is a tensor product and ##x## and ##y## are not matrices in coordinate representation. Can someone explain this?
 
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  • #2
To explain this we need more context, i.e., which concrete example are you discussing?
 
  • #3
LagrangeEuler said:
##x## and ##y## are not matrices in coordinate representation.
But they are matrices in any other representation, e.g. momentum representation. And even in coordinate representation they can be viewed as matrices, but in this representation calculations can be simplified by viewing them as ordinary numbers.
 
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  • #4
vanhees71 said:
To explain this we need more context, i.e., which concrete example are you discussing?
For instance 2d linear harmonic oscillator with this perturbation.
 
  • #5
I'm confused by your confusion.

A 1d quantum system "lives" in a Hilbert space ##\mathcal{H}_{x}## that is spanned by linear combination of base kets of the form ##\left|x\right\rangle##. You can also have another independent degree of freedom living in a Hilbert space ##\mathcal{H}_{y} ## with base kets of the form ##\left|y\right\rangle ##. Then, you can combine both degrees of freedom into a Hilbert space ##\mathcal{H}_{xy}=\mathcal{H}_{x}\otimes\mathcal{H}_{y}##, that is how the Hilbert space of a 2d quantum system is build. The base kets are of the form ##\left|xy\right\rangle \equiv\left|x\right\rangle \otimes\left|y\right\rangle ##

So ##V'=\alpha xy## is always a shorthand for ##V'=\alpha x \otimes y##, because that's how the operators acting on the Hilbert space of 2d quantum systems are build, by definition.

See Cohen-Tannoudji, quantum mechanics, vol 1, page 160.
 
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1. What is the tensor product in Cartesian coordinates?

The tensor product in Cartesian coordinates is a mathematical operation that combines two vectors or tensors to create a new tensor. It is commonly used in physics and engineering to describe the relationship between different physical quantities.

2. How is the tensor product calculated in Cartesian coordinates?

The tensor product in Cartesian coordinates is calculated by taking the outer product of the two vectors or tensors. This involves multiplying each element of one vector or tensor by each element of the other and arranging the results in a matrix.

3. What is the significance of the tensor product in Cartesian coordinates?

The tensor product in Cartesian coordinates is significant because it allows for the representation of complex relationships between physical quantities in a simple and concise manner. It also allows for the transformation of vectors and tensors between different coordinate systems.

4. Can the tensor product be applied to more than two vectors or tensors?

Yes, the tensor product can be applied to any number of vectors or tensors. The resulting tensor will have a higher order, with the number of indices equal to the number of vectors or tensors involved.

5. How is the tensor product related to other mathematical operations?

The tensor product is related to other mathematical operations, such as the dot product and cross product, as it involves the multiplication of two vectors or tensors. However, it is a more general operation that can be applied to any number of vectors or tensors and produces a tensor as the result.

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