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Schrödinger equation for a one dimensional particle by

Ciro Santilli 37 Updated 2025-07-16

We select for the general Equation "Schrodinger equation":

$x = x$ , the linear cartesian coordinate in the x direction
$\hat{H} = - \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x, t)$ , which analogous to the sum of kinetic and potential energy in classical mechanics

giving the full explicit partial differential equation:

i ℏ \frac{\partial ψ ( x , t )}{\partial t} = [- \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x, t)] ψ (x, t)

Equation 1.

Schrödinger equation for a one dimensional particle

The corresponding time-independent Schrödinger equation for this equation is:

[- \frac{ℏ ^{2}}{2 m} \frac{\partial ^{2}}{\partial x ^{2}} + V (x)] ψ (x) = E ψ (x)

(2)

Equation 2.

time-independent Schrödinger equation for a one dimensional particle

Schrödinger equation for a free one dimensional particle by

Ciro Santilli 37 Updated 2025-07-16

Schrödinger equation for a one dimensional particle with

V = 0

. The first step is to calculate the time-independent Schrödinger equation for a free one dimensional particle

Then, for each energy

E

, from the discussion at Section "Solving the Schrodinger equation with the time-independent Schrödinger equation", the solution is:

ψ (x) = \int_{E = - \infty}^{\infty} e^{\frac{2 m E i x}{ℏ}} e^{- i Et /ℏ} = e^{i \frac{2 m E x - Et}{ℏ}}

Therefore, we see that the solution is made up of infinitely many plane wave functions.

Plane wave function by

Ciro Santilli 37 Updated 2025-07-16

In this solution of the Schrödinger equation, by the uncertainty principle, position is completely unknown (the particle could be anywhere in space), and momentum (and therefore, energy) is perfectly known.

The plane wave function appears for example in the solution of the Schrödinger equation for a free one dimensional particle. This makes sense, because when solving with the time-independent Schrödinger equation, we do separation of variable on fixed energy levels explicitly, and the plane wave solutions are exactly fixed energy level ones.

Time-independent Schrödinger equation for a free one dimensional particle by

Ciro Santilli 37 Updated 2025-07-16

\frac{\partial ^{2}}{\partial x ^{2}} ψ (x) = - \frac{2 m E}{ℏ ^{2}} ψ (x)

so the solution is:

ψ (x) = e^{\frac{2 m E i x}{ℏ}}

(2)

We notice that the solution has continuous spectrum, since any value of

E

can provide a solution.

Quantum LC circuit by

Ciro Santilli 37 Updated 2025-07-16

A quantum version of the LC circuit!

TODO are there experiments, or just theoretical?

Hermite polynomials by

Ciro Santilli 37 Updated 2025-07-16

Show up in the solution of the quantum harmonic oscillator after separation of variables leading into the time-independent Schrödinger equation, much like solving partial differential equations with the Fourier series.

I.e.: they are both:

solutions to the time-independent Schrödinger equation for the quantum harmonic oscillator
a complete basis of that space

Hermite functions by

Ciro Santilli 37 Updated 2025-07-16

Not the same as Hermite polynomials.

Ladder operator by

Ciro Santilli 37 Updated 2025-07-16

www.physics.udel.edu/~jim/PHYS424_17F/Class%20Notes/Class_5.pdf by James MacDonald shows it well.

The operators are a natural guess on the lines of "if p and x were integers".

And then we can prove the ladder properties easily.

The commutator appear in the middle of this analysis.

Quantum tunnelling by

Ciro Santilli 37 Updated 2025-07-16

Examples:

flash memory uses quantum tunneling as the basis for setting and resetting bits
alpha decay is understood as a quantum tunneling effect in the nucleus

Atomic orbital by

Ciro Santilli 37 Updated 2025-07-16

In the case of the Schrödinger equation solution for the hydrogen atom, each orbital is one eigenvector of the solution.

Remember from time-independent Schrödinger equation that the final solution is just the weighted sum of the eigenvector decomposition of the initial state, analogously to solving partial differential equations with the Fourier series.

This is the table that you should have in mind to visualize them: en.wikipedia.org/w/index.php?title=Atomic_orbital&oldid=1022865014#Orbitals_table

Principal quantum number by

Ciro Santilli 37 Updated 2025-07-16

Determines energy. This comes out directly from the resolution of the Schrödinger equation solution for the hydrogen atom where we have to set some arbitrary values of energy by separation of variables just like we have to set some arbitrary numbers when solving partial differential equations with the Fourier series. We then just happen to see that only certain integer values are possible to satisfy the equations.

Azimuthal quantum number by

Ciro Santilli 37 Updated 2025-07-16

Fixed total angular momentum.

The direction however is not specified by this number.

To determine the quantum angular momentum, we need the magnetic quantum number, which then selects which orbital exactly we are talking about.

p-orbital by

Ciro Santilli 37 Updated 2025-07-16

d-orbital by

Ciro Santilli 37 Updated 2025-07-16

Magnetic quantum number by

Ciro Santilli 37 Updated 2025-07-16

Fixed quantum angular momentum in a given direction.

Can range between

\pm l

E.g. consider gallium which is 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p1:

the electrons in s-orbitals such as 1s, 2d, and 3d are $l = 0$ , and so the only value for $m_{l}$ is 0
the electrons in p-orbitals such as 2p, 3p and 4p are $l = 1$ , and so the possible values for $m_{l}$ are -1, 0 and 1
the electrons in d-orbitals such as 2d are $l = 2$ , and so the possible values for $m_{l}$ are -2, -1, 0 and 1 and 2

The z component of the quantum angular momentum is simply:

L_{z} = m_{l} ℏ

so e.g. again for gallium:

s-orbitals: necessarily have 0 z angular momentum
p-orbitals: have either 0, $- ℏ$ or $+ ℏ$ z angular momentum

Note that this direction is arbitrary, since for a fixed azimuthal quantum number (and therefore fixed total angular momentum), we can only know one direction for sure.

z

is normally used by convention.

Spin quantum number by

Ciro Santilli 37 Updated 2025-07-16

Spectroscopic notation by

Ciro Santilli 37 Updated 2025-07-16

This notation is cool as it gives the spin quantum number, which is important e.g. when talking about hyperfine structure.

But it is a bit crap that the spin is not given simply as

\pm 1/2

but rather mixes up both the azimuthal quantum number and spin. What is the reason?

Video 1.

Spectroscopic notation by Andre K (2014)

Source.

Solutions for the Schrodinger equation with multiple particles by

Ciro Santilli 37 Updated 2025-07-16

Bibliography:

Quantum Mechanics for Engineers by Leon van Dommelen (2011) "5. Multiple-Particle Systems"

Hartree-Fock method by

Ciro Santilli 37 Updated 2025-07-16