Inner and Outer Product of the Wavefunctions

• I
• Dario56
In summary, the 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 defined as $\langle v|w\rangle = v_1w_1 + v_2w_2 + ...+v_nw_n$, where $v_1,w_1, v_2,w_2...v_n,w_n$ are the corresponding components of both vectors in a particular basis set. In quantum mechanics, wavefunctions are vectors in the Hilbert space, and we can express them as a linear combination of a complete and orthonormal basis set
Dario56
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.

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Dario56 said:
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: $$\Psi = \sum_k c_k\psi_k$$

We can express the last equation in the vector form (Dirac notation) where we write it as the inner product: $$\Psi = \langle c|\psi\rangle = c^\dagger \psi$$

That's not correct. You are confusing vectors and scalars.

The equivalent of $$\Psi = \sum_k c_k\psi_k$$ in Dirac notation is $$\ket{\Psi} = \sum_k c_k \ket{\psi_k}$$

Dario56 said:
If we have two wavefunctions, ##\Psi_1## and ##\Psi_2## than their inner product is defined as: $$\langle \Psi_1|\Psi_2\rangle = \sum_{i,j} c_i^*c_j \langle \psi_i|\psi_j\rangle$$
where we expanded wavefunctions in some basis set.
Your notation is sloppy here and causes problem later. ##\ket{\Psi_1}## and ##\ket{\Psi_2}## being distinct, you cannot use the same coefficients ##c## for both. So instead of ##c_i^*c_j##, you should have. something like ##c_{1,i}^*c_{2,j}##.
Dario56 said:
$$\langle \Psi_1|\Psi_2\rangle = \sum_{i,j} c_i^*c_j \delta_{ij} = \sum_i |c_i|^2$$
$$\langle \Psi_1|\Psi_2\rangle = \sum_{i,j} c_{1,i}^*c_{2,j} \delta_{ij} = \sum_i c_{1,i}^*c_{2,i}$$
Dario56 said:
Outer product of the wavefunction ##\Psi## with itself can be defined as: $$|\Psi\rangle \langle \Psi| = \sum_k c_k |\psi_k\rangle c_k^* \langle \psi_k| = \sum_k |c_k|^2 |\psi_k\rangle \langle \psi_k|$$

where we expressed the wavefunction in the basis set.
No, it is a double sum
$$|\Psi\rangle \langle \Psi| = \sum_{i,j} c_i |\psi_i\rangle c_j^* \langle \psi_j|$$
The outer product is an operator (equivalent to a matrix, not a vector).

PeroK, topsquark, malawi_glenn and 1 other person
DrClaude said:
That's not correct. You are confusing vectors and scalars.

The equivalent of $$\Psi = \sum_k c_k\psi_k$$ in Dirac notation is $$\ket{\Psi} = \sum_k c_k \ket{\psi_k}$$Your notation is sloppy here and causes problem later. ##\ket{\Psi_1}## and ##\ket{\Psi_2}## being distinct, you cannot use the same coefficients ##c## for both. So instead of ##c_i^*c_j##, you should have. something like ##c_{1,i}^*c_{2,j}##.

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

No, it is a double sum
$$|\Psi\rangle \langle \Psi| = \sum_{i,j} c_i |\psi_i\rangle c_j^* \langle \psi_j|$$
The outer product is an operator (equivalent to a matrix, not a vector).

That's not correct. You are confusing vectors and scalars.

The equivalent of $$\Psi = \sum_k c_k\psi_k$$ in Dirac notation is $$\ket{\Psi} = \sum_k c_k \ket{\psi_k}$$
Yes, I made a mistake here. I corrected this.

Your notation is sloppy here and causes problem later. ##\ket{\Psi_1}## and ##\ket{\Psi_2}## being distinct, you cannot use the same coefficients ##c## for both. So instead of ##c_i^*c_j##, you should have. something like ##c_{1,i}^*c_{2,j}##.
Yup, this is indeed not correct. Will correct it. It is easy to make a mistake in the notation when you write in very general terms.

No, it is a double sum
$$|\Psi\rangle \langle \Psi| = \sum_{i,j} c_i |\psi_i\rangle c_j^* \langle \psi_j|$$
The outer product is an operator (equivalent to a matrix, not a vector).

Double sum is generally correct. However, since I have the same wavefunction ##\Psi## is it reasonable to expand it in the same basis set which eliminates the need to write the double sum?

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Dario56 said:
Double sum is generally correct. However, since I have the same wavefunction ##\Psi## is it reasonable to expand it in the same basis set which eliminates the need to write the double sum?
It is expanded in the same basis here, namely ##\ket{\psi}##. There is no further simplification possible.

topsquark
DrClaude said:
It is expanded in the same basis here, namely ##\ket{\psi}##. There is no further simplification possible.
Oh, yes. That is correct. Now, when all mistakes are corrected, we can get to my original question which is: How to calculate outer product of two basis functions? $$|\psi_i\rangle\langle\psi_j|$$

As I said in the main post, inner product of the mentioned functions is defined as: $$\langle \psi_i |\psi_j \rangle = \int \psi_i^* \psi_j \,dx$$

I can't really find what is the general definition for the outer product of two functions, though.

It's an operator. It transforms a state into another state. How it will act specifically depends on the representation.

Dario56 and topsquark
Dario56 said:
...I can't really find what is the general definition for the outer product of two functions, though.
If you are looking for something like integral form of finding the inner product, you aren't going to get anywhere. Like DrClaude said, the outer product is going to be a matrix, or operator.For example, in the bra-ket notation, let's look at the spin 1/2 system. Let's write + and - state in the "usual" basis:
##|+> = \left ( \begin{matrix} 1 \\ 0 \end{matrix} \right )##

##|-> = \left ( \begin{matrix} 0 \\ 1 \end{matrix} \right )##

Then we can get things like the outer product
##|+><-| = \left ( \begin{matrix} 1 \\ 0 \end{matrix} \right ) \left ( \begin{matrix} 0 & 1 \end{matrix} \right ) = \left ( \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right )##

The exact form of the outer product depends on the basis we use to define the wavefunctions.

-Dan

Dario56
topsquark said:
If you are looking for something like integral form of finding the inner product, you aren't going to get anywhere. Like DrClaude said, the outer product is going to be a matrix, or operator.For example, in the bra-ket notation, let's look at the spin 1/2 system. Let's write + and - state in the "usual" basis:
##|+> = \left ( \begin{matrix} 1 \\ 0 \end{matrix} \right )##

##|-> = \left ( \begin{matrix} 0 \\ 1 \end{matrix} \right )##

Then we can get things like the outer product
##|+><-| = \left ( \begin{matrix} 1 \\ 0 \end{matrix} \right ) \left ( \begin{matrix} 0 & 1 \end{matrix} \right ) = \left ( \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right )##

The exact form of the outer product depends on the basis we use to define the wavefunctions.

-Dan
Yes, so outer product can't be defined for functions in the same way as the inner product can. We can only define it, if we represent quantum states with vectors in certain basis like you did. Since quantum states are vectors in the Hilbert space, this isn't a problem.

Of course, you can write out what the projector looks like when applied to a wave function:
$$\ket{\phi} \braket{\phi | \Psi} \doteq \phi \int \phi^* \Psi dx$$

vanhees71, topsquark and Dario56

1. What is the difference between the inner and outer product of wavefunctions?

The inner product of two wavefunctions is a mathematical operation that results in a scalar value, while the outer product results in a matrix. The inner product is used to determine the overlap or similarity between two wavefunctions, while the outer product is used to combine two wavefunctions to create a new one.

2. How is the inner product of wavefunctions calculated?

The inner product is calculated by integrating the product of two wavefunctions over the entire space. This can be represented as ∫Ψ₁(x)Ψ₂(x)dx, where Ψ₁ and Ψ₂ are the two wavefunctions being multiplied.

3. What is the significance of the inner product in quantum mechanics?

The inner product is an important tool in quantum mechanics as it allows us to calculate the probability of a particle being in a certain state. The square of the inner product gives us the probability amplitude, which is used to determine the probability of finding a particle in a specific location or state.

4. Can the inner and outer product of wavefunctions be used for any type of wavefunction?

Yes, the inner and outer product can be used for any type of wavefunction, including complex-valued wavefunctions. However, the resulting values may also be complex numbers.

5. How are the inner and outer product of wavefunctions related to each other?

The outer product can be seen as the generalization of the inner product. It is used to combine two wavefunctions, while the inner product is used to determine the overlap between two wavefunctions. Additionally, the outer product can be calculated using the inner product, as it is represented by the outer product of the two wavefunctions' ket and bra vectors.

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