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My understanding is that in quantum mechanics the wavefunction may be expressed as a function or as a ket vector (composed of many orthogonal ket vectors). I'm not too sure about the further differences between these two approaches. For example, I am familiar with operating on the wavefunction as a function with the hamiltonian operator, but when you use ket vectors can you use the hamiltonian operator in the same way or do you have to find matrix representations of the operators?

I would also be grateful for any links to websites or books explaining the different approaches.

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The "wavefunction" way of talking about things is a special case of the more abstract "Hilbert space" formulation.

The abstract formulation says that states live in a Hilbert space, that is a complex vector space with an inner product (plus some technical assumption about completeness). The Hamiltonian is then a linear operator on that space. The way you define your Hilbert space will affect what your operators look like and how you define them.

Perhaps the most important example of an inner product space is $L^2$, the space of all square-integrable functions (that is, $\psi:\mathbb{R}\to\mathbb{C}$ such that $\int_{-\infty}^\infty |\psi(x)|^2dx$ exists), with inner product $\langle\phi,\psi\rangle = \int_{-\infty}^\infty \phi(x)^*\psi(x)dx$. This is just what you'd normally talk about with normalizable wavefunctions. Natural operators can be formed out of things like multiplication by functions, and derivatives.

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