Understanding Basis Change with Hamiltonian Matrices

In summary: This is another way of saying that ##SHS^{-1}## is a representation of ##H## in the primed basis.In summary, we are given two vectors, la> = (1,0) and lb> = (0,1), and a Hamiltonian H which is a 2x2 matrix with 2 on the diagonal entries and zero elsewhere. We are then asked to represent H in the basis of the vectors la'> = 1/sqrt(2)(1,1) and lb'> = 1/sqrt(2)(1,-1), which are also eigenvectors of H. This means finding the components of H in the new basis using the basis change equation H' = SHS
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We are given the vectors la> = (1,0) and lb> = (0,1) and then a Hamiltonian H which is a 2x2 matrix with 2 on the diagonal entires and zero elsewhere. I am asked to now represent H in the basis of the vectors la'> = 1/sqrt(2)(1,1) and lb'> = 1/sqrt(2)(1,-1), which are also eigenvectors of H because of the degeneracy of the eigenvalues of H.
I always get confused with these basis-change problems. First of all: What does it even mean that a matrix is represented in a particular basis? Why is H necessarily represented in the basis (1,0) and (0,1)?
Secondly I know that the basis change equation is:
H' = SHS^-1 (1)
But my QM also tells me that the elements of H' can be found by taking the inner products <a'lHla'>, <a'lHlb'> etc. How is this equivalent to (1)?
 
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aaaa202 said:
What does it even mean that a matrix is represented in a particular basis?
See this FAQ post: https://www.physicsforums.com/threads/matrix-representations-of-linear-transformations.694922/ [Broken]
 
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Okay but I am still not sure why for an orthonormal basis the basis change equation:
B = SAS^-1
Reduces to just calculating B using the equations B_ij = <bil A lbj> , where b1,b2,b3... are the new basis vectors of the new basis expressed in the old basis.
 
  • #4
I didn't have time to answer that before. Let U be the linear operator that takes the old basis vectors to the new, i.e. ##e_i'=Ue_i##. I'll start by proving that this U must be unitary (assuming that both bases are orthonormal). For all i,j, we have
$$\langle e_i,e_j\rangle =\delta_{ij}=\langle e_i',e_j'\rangle =\langle Ue_i,Ue_j\rangle =\langle e_i,U^\dagger U e_j\rangle,$$ and therefore
$$\langle e_i,(I-U^\dagger U)e_j\rangle =0.$$ This implies that for all j and all vectors x, we have
$$\langle x,(I-U^\dagger U)e_j\rangle =\left\langle\sum_{i=1}^2 x_i e_i,(I-U^\dagger U)e_j\right\rangle =\sum_{i=1}^2 (x_i)^* \langle e_i,(I-U^\dagger U)e_j\rangle =0.$$ In particular, this holds when ##x=(I-U^\dagger U)e_j##. That implies that for all j, we have ##(I-U^\dagger U)e_j=0##. This implies that ##U^\dagger U=I##.

Now if we define ##S=U^{-1}## and use that ##U^\dagger=U^{-1}##, we have
\begin{align}
H'_{ij}=\langle e_i',He_j'\rangle = \langle Ue_i,HUe_j\rangle =\langle e_i,U^\dagger HUe_j\rangle =(U^\dagger HU)_{ij} =(SHS^{-1})_{ij}
\end{align} In other words, the components of ##H## in the primed basis are the same as the components of ##SHS^{-1}## in the unprimed basis.
 
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I can understand your confusion about basis change problems. Let me explain the concept of basis change and how it relates to Hamiltonian matrices.

A basis is a set of linearly independent vectors that span a vector space. In other words, any vector in that space can be written as a linear combination of the basis vectors. In the given problem, the basis vectors are la> = (1,0) and lb> = (0,1). These are the standard basis vectors for a 2-dimensional space.

Now, a matrix represents a linear transformation. In this case, the Hamiltonian matrix H represents a linear transformation in the 2-dimensional space spanned by la> and lb>. This means that H takes any vector in this space and transforms it into another vector in the same space.

The matrix representation of a linear transformation depends on the basis used to represent the vectors. In other words, the same linear transformation can have different matrix representations depending on the basis chosen. This is where basis change comes in.

In your problem, you are asked to represent H in a new basis, which is given by la'> = 1/sqrt(2)(1,1) and lb'> = 1/sqrt(2)(1,-1). This means that we want to find the matrix representation of H in this new basis.

To do this, we use the basis change equation (1) that you have mentioned. Here, S is the matrix whose columns are the new basis vectors la'> and lb'>, and S^-1 is its inverse. This equation essentially transforms the matrix H from the old basis to the new basis.

Now, to answer your question about why H is necessarily represented in the basis (1,0) and (0,1), it is because this is the standard basis for a 2-dimensional space. Any vector or matrix in this space can be represented in this basis. However, when we want to represent the same vector or matrix in a different basis, we use the basis change equation.

Finally, you are correct in saying that the elements of H' can be found by taking the inner products <a'lHla'>, <a'lHlb'> etc. This is equivalent to the basis change equation (1) because the inner product essentially represents the transformation of a vector from one basis to another.

I hope this explanation helps you understand the concept of basis change and its relation to Hamiltonian matrices. It is
 

What is a Hamiltonian matrix?

A Hamiltonian matrix is a square matrix that represents the total energy of a physical system in quantum mechanics. It is named after the physicist William Rowan Hamilton and is commonly used to study and understand the dynamics of quantum systems.

What is basis change in relation to Hamiltonian matrices?

Basis change refers to the transformation of a Hamiltonian matrix from one basis (set of vectors) to another. This is often done to simplify calculations or to better understand the system's behavior in a different basis.

Why is understanding basis change important for studying Hamiltonian matrices?

Understanding basis change is important because it allows for a more intuitive and efficient interpretation of the Hamiltonian matrix. It also allows for the identification of important features and relationships within the matrix that may not be apparent in its original basis.

What are the mathematical techniques used to perform basis change on Hamiltonian matrices?

The most commonly used techniques for basis change on Hamiltonian matrices include diagonalization, unitary similarity transformations, and canonical transformations. These methods involve manipulating the matrix using specific operations to transform it into a different basis.

How does basis change affect the interpretation of Hamiltonian matrices?

Basis change affects the interpretation of Hamiltonian matrices by revealing new insights and relationships within the matrix. It can also simplify calculations and make it easier to visualize the dynamics of the system. Additionally, basis change can help identify symmetries and other important features of the matrix that may not be apparent in its original form.

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