Quantum mechanics matrices

In summary, the conversation discusses the confusion and complexity surrounding undergraduate quantum mechanics. The wave function is an abstract vector in an infinite dimensional Hilbert space and is represented as superpositions of two states. The Hamiltonian is represented as a 4x4 matrix and the state by a 2x2 matrix. The conversation also touches on the use of linear algebra and Fourier analysis in understanding QM and how it is taught in undergraduate courses. The actual Schroedinger equation is also discussed.
  • #1
cooev769
114
0
Just trying to get my head around undergraduate quantum mechanics and they throw a lot of stuff at you, so some help is appreciated.

I understand that the wave function is some abstract vector living in an infinite dimensional hilbert space, and that it's a function. But then the textbook I'm reading goes on to represent the state as superpositions of two states:

s=al1>+bl2>

I've actually done quite a lot of QM calculations and am very familiar with the maths involved in ket-bra notation and what they represent. But I'm just confused as to how they tie in with the Hilbert space and infinite dimensional vectors and such. They've now represented the hamiltonian as a 4x4 matrix, the state by 2/2 matrix and I'm just confused as to how the functions tie into this at all. Do these states have anything to do with functions, or the wave function. I mean we were specifically told not to think of the wave function or QM in matrix form and just think of it as an abstract vector function, but now all the math we are doing is using superpositions and using vectors to calculate eigenvectors and eigenfunctions of the hamiltonian and so on. It's just all very confusing and overwhelming haha.

Thanks for any help.
 
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  • #2
Are these l1> and l2> just arbitrary kets, which can represent any function? Because in this case l1> is a ket vector (1 0) and l2> is a ket vector (0 1) so they're linearly independent. But how do these vectors have anything to do with functions for example if I wanted to imagine this in terms of the wave function.
 
  • #3
In abstract form the Shroedinger equation is H |psi> = E |psi>, where H is the Hamiltonian operator, |psi> is a state vector, and E is the energy associated with that state vector.

This is in the form of an eigenvalue equation (from linear algebra) ... since H is a linear operator it also has a matrix representation. The dimensionality of the matrix for H corresponds to the dimensionality of the state space for the eigenvectors (including any degeneracy), and the number of eigenvalues.

Especially for infinite dimensional cases it is useful to think of the vector space as a space of functions ... then Fourier analysis is the tool of choice. The Hilbert space is often viewed as a function space: the vectors are linear combinations of functions.

But most undergraduate QM courses don't discuss it in such abstract terms because ... then the course would be an advanced course in linear algebra and Fourier theory. Instead they just teach you the mechanics of what you need, and show you how to setup and solve problems.

Much like introductory calculus. You don't really find out what is going on unless you are a math major, or a brave soul who stumbles into a course on real analysis.
 
  • #4
I would like to point out that the actual Schroedinger equation is $$\hat{H}\left|\Psi\right>=i\hbar\frac{\partial}{\partial t}\left|\Psi\right>$$

The equation that PED gave is the time-independent Schroedinger equation valid for time-independent Hamiltonians for which one uses separation of variables on the original PDE to turn the Schroedinger equation into a ODE.
 

1. What are matrices in quantum mechanics?

Matrices in quantum mechanics are mathematical representations of physical quantities, such as energy, momentum, and spin, in the quantum world. They are used to describe the state of a quantum system and the evolution of that system over time.

2. How are matrices used in quantum mechanics?

Matrices are used in quantum mechanics to represent observables, which are measurable properties of a quantum system. They are also used to calculate the probabilities of different outcomes for a given measurement, as well as to describe the mathematical operations that occur during a quantum measurement.

3. What is the significance of matrix multiplication in quantum mechanics?

Matrix multiplication is significant in quantum mechanics because it allows for the calculation of the probabilities of different outcomes for a given measurement. It also helps to describe the evolution of a quantum system over time, as quantum states can be represented as matrices and can be multiplied together to determine the state at a later time.

4. How do matrices and operators relate to each other in quantum mechanics?

In quantum mechanics, operators are used to represent physical quantities, such as energy or momentum, and can be written as matrices. These operators act on quantum states, causing them to change or evolve over time. Matrices and operators are closely related and are essential tools for understanding the behavior of quantum systems.

5. Can matrices be used to solve problems in quantum mechanics?

Yes, matrices are an essential tool for solving problems in quantum mechanics. They are used to calculate the probabilities of different outcomes for a given measurement and to describe the evolution of quantum states over time. Matrices are also used to represent complex quantum systems, making them a crucial component of quantum mechanics research and problem-solving.

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