Quantenphysik 🎲

They show the likelihood of finding the electron in a given region of space around the nucleus, depending on the quantum numbers n, l, and m. The wave functions are solutions to the Schrödinger equation for the hydrogen atom, and they depend on three variables: r, θ, and φ, which are the spherical coordinates of the electron. The eight stationary states shown in the image are: The ground state (n = 1, l = 0, m = 0), which has a spherical shape and is symmetric around the nucleus. The probability density is highest at the nucleus and decreases exponentially with increasing r. The first excited state (n = 2, l = 0, m = 0), which has a spherical shape and is symmetric around the nucleus. The probability density has a node at r = 2a0, where a0 is the Bohr radius, and peaks at r = a0 and r = 4a0. The second excited state (n = 2, l = 1, m = 0), which has a dumbbell shape and is symmetric along the z-axis. The probability density has a node at the nucleus and peaks at r = a0 for θ = π/2. The third excited state (n = 2, l = 1, m = ±1), which has a dumbbell shape and is symmetric along the x-axis or y-axis, depending on the sign of m. The probability density has a node at the nucleus and peaks at r = a0 for θ = π/4 or θ = 3π/4. The fourth excited state (n = 3, l = 0, m = 0), which has a spherical shape and is symmetric around the nucleus. The probability density has two nodes at r ≈ 1.6a0 and r ≈ 4.8a0, and peaks at r ≈ a0, r ≈ 3a0, and r ≈ 6a0. The fifth excited state (n = 3, l = 1, m = 0), which has a dumbbell shape and is symmetric along the z-axis. The probability density has two nodes at r ≈ 2.5a0 and r ≈ 5a0 for θ = π/2, and peaks at r ≈ a0 and r ≈ 4a0 for θ = π/2. The sixth excited state (n = 3, l = 1, m = ±1), which has a dumbbell shape and is symmetric along the x-axis or y-axis, depending on the sign of m. The probability density has two nodes at r ≈ 2.5a0 and r ≈ 5a0 for θ = π/4 or θ = 3π/4, and peaks at r ≈ a0 and r ≈ 4a0 for θ = π/4 or θ = 3π/4. The seventh excited state (n = 3, l = 2, m = 0), which has a cloverleaf shape and is symmetric along the z-axis. The probability density has three nodes: one at the nucleus, one at r ≈ a0 for θ ≈ π/6 or θ ≈ 5π/6, and one at r ≈ a0 for θ ≈ π/2. It peaks at r ≈ a0 for θ ≈ π/3 or θ ≈ 2π/3. The image also shows how the position probability distributions are projected along the y-axis by integrating over x and z. This gives an idea of how the electron cloud appears when viewed from above or below.