- #1

Dario56

- 289

- 44

Inner product is a generalization of the dot product on spaces other than Euclidean and for vectors it is defined in the same way as the dot product. If we have two vectors $v$ and $w$, than their inner product is: $$\langle v|w\rangle = v_1w_1 + v_2w_2 + ...+v_nw_n $$

where $v_1,w_1, v_2,w_2...v_n,w_n$ are the corresponding components of both vectors in particular basis set.

We know that wavefunctions are vectors in the Hilbert space. This comes from the linear properties of quantum systems (Schrödinger equation). Namely, we can express every wavefunction as the linear combination of particular complete and orthonormal basis set: $$ |\Psi\rangle = \sum_k c_k|\psi_k\rangle$$If we have two wavefunctions, $\Psi_1$ and $\Psi_2$ than their inner product is defined as: $$ \langle \Psi_1|\Psi_2\rangle = \sum_{i,j} c_{1,i}^*c_{2,j} \langle \psi_i|\psi_j\rangle $$

where we expanded wavefunctions in some basis set.

To calculate the inner product, we need to define what it means to take the inner product between functions generally and we need to see how it applies to the basis functions. Inner product between two functions in general $\psi_i$ and $\psi_j$ is defined differently compared to vectors although definition is similar: $$ \langle\psi_i|\psi_j\rangle = \int\psi_i^* \psi_j \,dx $$

Idea is that we multiply both functions for every value of $x$ and than sum over all possible values of $x$. Since $x$ is continuous, we integrate rather than sum.

If the basis set is orthonormal than inner product of two basis functions in the Hilbert space must be equal to Kronecker delta, $\delta_{ij}$ in the similar way as it is the case for orthonormal vectors in Euclidean space (dot product): $$ \langle\psi_i|\psi_j\rangle = \int\psi_i^* \psi_j \,dx = \delta_{ij}$$

$$ \langle \Psi_1|\Psi_2\rangle = \sum_{i,j} c_{1,i}^*c_{2,j} \delta_{ij} = \sum_i c_{1,i}c_{2,i}$$

Outer product of the wavefunction $\Psi$ with itself can be defined as: $$ |\Psi\rangle \langle \Psi| = \sum_{i.j} c_i |\psi_i\rangle c_j^* \langle \psi_j| = \sum_{i,j} c_ic_j^* |\psi_i\rangle \langle \psi_j| $$

where we expressed the wavefunction in the basis set.

I am not sure how to calculate the outer product between the basis functions? For the inner product, I know how is it defined generally and I know it must be equal to Kronecker delta in the case of basis functions.

I also know that sum of the outer products of basis functions should be equal to the identity, but I don't know how to define and calculate it in the case of individual basis functions.

functions.

where $v_1,w_1, v_2,w_2...v_n,w_n$ are the corresponding components of both vectors in particular basis set.

We know that wavefunctions are vectors in the Hilbert space. This comes from the linear properties of quantum systems (Schrödinger equation). Namely, we can express every wavefunction as the linear combination of particular complete and orthonormal basis set: $$ |\Psi\rangle = \sum_k c_k|\psi_k\rangle$$If we have two wavefunctions, $\Psi_1$ and $\Psi_2$ than their inner product is defined as: $$ \langle \Psi_1|\Psi_2\rangle = \sum_{i,j} c_{1,i}^*c_{2,j} \langle \psi_i|\psi_j\rangle $$

where we expanded wavefunctions in some basis set.

To calculate the inner product, we need to define what it means to take the inner product between functions generally and we need to see how it applies to the basis functions. Inner product between two functions in general $\psi_i$ and $\psi_j$ is defined differently compared to vectors although definition is similar: $$ \langle\psi_i|\psi_j\rangle = \int\psi_i^* \psi_j \,dx $$

Idea is that we multiply both functions for every value of $x$ and than sum over all possible values of $x$. Since $x$ is continuous, we integrate rather than sum.

If the basis set is orthonormal than inner product of two basis functions in the Hilbert space must be equal to Kronecker delta, $\delta_{ij}$ in the similar way as it is the case for orthonormal vectors in Euclidean space (dot product): $$ \langle\psi_i|\psi_j\rangle = \int\psi_i^* \psi_j \,dx = \delta_{ij}$$

$$ \langle \Psi_1|\Psi_2\rangle = \sum_{i,j} c_{1,i}^*c_{2,j} \delta_{ij} = \sum_i c_{1,i}c_{2,i}$$

Outer product of the wavefunction $\Psi$ with itself can be defined as: $$ |\Psi\rangle \langle \Psi| = \sum_{i.j} c_i |\psi_i\rangle c_j^* \langle \psi_j| = \sum_{i,j} c_ic_j^* |\psi_i\rangle \langle \psi_j| $$

where we expressed the wavefunction in the basis set.

I am not sure how to calculate the outer product between the basis functions? For the inner product, I know how is it defined generally and I know it must be equal to Kronecker delta in the case of basis functions.

I also know that sum of the outer products of basis functions should be equal to the identity, but I don't know how to define and calculate it in the case of individual basis functions.

functions.

Last edited: