u/TROSE9025

Although modern quantum mechanics often prefers an algebraic approach, wavefunction-based quantum mechanics still remains the main approach in undergraduate courses.

I believe the algebraic approach should be central, but students also need enough practice with the wavefunction-based Schrödinger equation.

This post carefully examines reflection and transmission for a particle in a rectangular barrier potential.

It also discusses probability conservation and special cases such as the condition for perfect transmission, with full mathematical steps and clear physical meaning.

I hope this helps.

This post is not about proof-based linear algebra in pure mathematics.
I would like to make it clear in advance that the scope here is limited to linear algebra as a bridge course for applications in finite-dimensional Hilbert spaces, such as quantum mechanics and quantum computing / quantum information theory.

If we have linearly independent vectors, the Gram–Schmidt process is a method for constructing new vectors that are mutually orthogonal while still spanning the same space.
In other words, it is a process for finding an easier basis to work with without changing the space itself.

I hope this helps.

The inner product is one of the most important ideas in linear algebra, especially in many applied fields.

It measures, in a broad sense, how much two vectors overlap.

Its meaning is interpreted a little differently depending on the field, but what is common is that it helps define the structure of a vector space.

In quantum mechanics, the inner product is closely connected to normalization, probability, and unitary transformations.

Here, I try to connect these ideas step by step through Dirac’s bra-ket notation, geometric meaning, and matrix representation.

By Taeryeon.