## Schrodinger Equation and The Wave Function

In this lesson you will learn about the time-dependent and independent Schrödinger equation (1d, 3d), how it is derived and what you can do with it.

Theoretical tools and models of quantum mechanics

Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.

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Lesson ## Schrodinger Equation and The Wave Function

In this lesson you will learn about the time-dependent and independent Schrödinger equation (1d, 3d), how it is derived and what you can do with it.

## Related formulas

Formula ## Time independent Schrödinger equation (3d)

`$$ W \, \mathit{\Psi} ~=~ - \frac{\hbar^2}{2\class{brown}{m}} \, \nabla^2 \, \mathit{\Psi} ~+~ W_{\text{pot}} \, \mathit{\Psi} $$`Formula ## Time-dependent Schrödinger equation (3d)

`$$ i \, \hbar \, \frac{\partial \mathit{\Psi}}{\partial t} ~=~ - \frac{\hbar^2}{2\class{brown}{m}} \, \nabla^2 \, \mathit{\Psi} ~+~ W_{\text{pot}} \, \mathit{\Psi} $$`## Related Illustrations

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Lesson ## Bra-Ket Notation

Here you will learn how Bra-Ket notation (Dirac notation) is defined, which computational rules exist for it and which advantages this notation brings.

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Lesson ## Hermitian Operators (Matrices) in Quantum Mechanics

Learn what Hermitian operators and matrices are and what three important properties they have. Examples are also made.

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Lesson ## Angular Momentum in Quantum Mechanics: Commutators and Eigenvalues

Here you will learn about angular momentum in quantum mechanics, its commutators, and how angular momentum states and eigenvalues are generated using ladder operators.

## Related formulas

Formula ## Angular momentum commutator (Lx and Ly)

`$$ [ L_{\text x}, L_{\text y} ] ~=~ \mathrm{i} \, \hbar \, L_{\text z} $$`Formula ## Commutator Between Angular Momentum Squared and Angular Momentum Component

`$$ [ L^2, \, L_j ] ~=~ 0 $$`## Related Illustrations

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Derivation ## Finite Potential Well: Wave Function and Energy

Derivation of allowed energies and associated wave functions inside and outside a one-dimensional finite potential box.

## Related Illustrations