Lecture 41 (Hydrogen Atom and Spatial Quantization) Physics Fall 2018 Douglas Fields

Transcription

1 Lecture 41 (Hydrogen Atom and Spatial Quantization) Physics Fall 2018 Douglas Fields

2 States A state of the electron is described by a specific wavefunction (or a specific combination of wavefunctions in the case of a superposition state). The wavefunction is the complete wavefunction: with, = 2r na (,,, ) (,, ) r t = r e And for states with determined energy (= eigenstates = stationary states), we solved the time independent 3D Schrodinger equation in spherical coordinates with the Coulomb potential to get: 0 3 ( n ) n( n + ) 2 1! / n m ( r,, ) = e L 1, l n Y na0 2! n = 1, 2,3, ml ( ) ( ) and, = 0,1, 2,, n 1 m iet l =,,

3 The Hydrogen Atom Note that we now have three integers that represent particular eigenstates of energy: n = 1, 2,3, = 0,1,2,, n 1 m =,, We should have expected three quantum numbers, since we are in three dimensions (like n x, n y, and n z for the 3D box). Here, n represents the primary quantum number, and the energy of the eigenstates are given by: E n 4 1 me 13.6 = 2n = ( 4 ) ev n Recognize this? The fact that the Schrödinger s equation gives us the same energy states as Bohr s model (which were experimentally verified) is good verification that it is good model. In fact, we will find out that it tells us something about the atomic orbitals that is in disagreement with Bohr s model!

4 Hydrogen Wave Functions The wave functions look like this, where we are looking at a crosssection of the 3-D wave functions, (to get the picture in your head, you must rotate these around the z-axis) and color is used to show the amplitude of the wave function as a function of position. Note that these are the wave functions for a single electron in different states, depending on the quantum numbers n, l, and m l. "Hydrogen Density Plots" by PoorLeno (talk) - the English language Wikipedia (log).original text: I created this work entirely by myself. References:Forinash, Kyle. Hydrogen W Simulation. Indiana University Southeast. Retrieved on Tokita, Sumio; Sugiyama, Takao; Noguchi, Fumio; Fujii, Hidehiko; Kobayashi, Hidehiko (2006). "An Attempt to Construct an Isosurface Having Symmetry Elements". Journal of Computer Chemistry, Japan 5 (3): DOI: /jccj Licensed under Public Domain via Commons -

5 Probability Distributions The probability of finding an electron within a radial shell dr is just given by: P( r) dr = dv = 4 r dr The factor dv goes to zero at r = 0, so that the probability distribution functions P(r), (shown in the figure here) go to zero even though the wave functions do not (see above). Note that there are nodes in the distributions, but whenever l is maximum for a given n, there is a single peak at n 2 a 0, the same result given by the Bohr model.

6 Angular Momentum Quantization So, what do the quantum numbers mean? The first, n, determines the average radial distance of the electron from the center of the nucleus, and is the primary driver of the electron s energy state. The second, l, determines the quantization of the orbital angular momentum: ( ) L= + 1 = 0,1, 2 n 1 Note that there are n different possibilities for the state of the orbital angular momentum for the n th energy level. Also note that when l=0, the magnitude of the orbital angular momentum is zero which is different from Bohr s model, where the electron is always orbiting around the nucleus. This means that for l=0 states, the electron spends a small, but not insignificant amount of time inside the nucleus.

7 Angular Momentum Quantization What about m l? The third quantum number defines the projection of the angular momentum L onto some axis (which we will denote by the z-axis). Note that in the absence of any probe, the potential is spherically symmetric, so we are free to define that in any direction. Lz = m m = 0, 1, 2 For an l=2 state, L z can take on 5 values, as shown in the figure.

8 Quantization and Mathematics The quantization of E, L and L z can be viewed as the result of the boundary conditions, normalization and periodicity of the wave functions. The condition that Θ(θ) does not go to infinity results in the quantum number l. The condition that Φ(φ) is periodic results in the quantum number m l. But one can also view the particular quantization rules in terms of the uncertainty principle: If any component of L could be equal to the magnitude, then we would have the electron orbiting in a plane such that p z would be zero. That absolute knowledge of one component of the momentum would require a complete uncertainty of its position in that coordinate. Note that L x and L y are undetermined up to some value, leading to a definite and reasonable uncertainty on position.

9 Quantum States So, the electron trajectories are defined by states, each state determined by the quantum numbers n, l, and m l. These states form the basis states of the differential equation (the Schrödinger s equation), which are then the complete set of solutions. Historically, these states have been given names spectroscopic notation.

10 Breaking Symmetry Remember that degeneracies were the reflection of symmetries. What if we break some of the symmetry by introducing something that distinguishes one direction from another? One way to do that is to introduce a magnetic field. To investigate what happens when we do this, we will briefly use the Bohr model in order to calculate the effect of a magnetic field on a orbiting electron. In the Bohr model, we can think of an electron causing a current loop. This current loop causes a magnetic moment: = IA We can calculate the current and the area from a classical picture of the electron in a circular orbit: ev 2 evr = = IA = r = 2 r 2 Where the minus sign just reflects the fact that it is negatively charged, so that the current is in the opposite direction of the electron s motion.

11 Bohr Magneton Now, we use the classical definition of angular momentum to put the magnetic moment in terms of the orbital angular momentum: evr = 2 L = mvr = e L 2m And now use the Bohr quantization for orbital angular momentum: L= n e B =, when n=1 2m Which is called the Bohr magneton, the magnetic moment of a n=1 electron in Bohr s model.

12 Back to Breaking Symmetry It turns out that the magnetic moment of an electron in the Schrödinger model has the same form as Bohr s model: e e z = Lz = ml 2m 2m Now, what happens again when we break a symmetry, say by adding an external magnetic field?

13 Back to Breaking Symmetry Thanks, yes, we should lose degeneracies. e e z = Lz = ml 2m 2m But how and why? Well, a magnetic moment in a magnetic field feels a torque: = B And thus, it can have a potential energy: Or, in our case: U = B e U = zb = ml B = ml BB 2m

14 Zeeman Effect This potential energy breaks the degeneracy and splits the degenerate energy levels into distinct energies. 4 1 me e E = n, m m l l B 4 2n + 2m ( ) This was first seen in 1896 by Pieter Zeeman 0

15 Zeeman Effect In 1896, three years after submitting his thesis on the Kerr effect, he disobeyed the direct orders of his supervisor and used laboratory equipment to measure the splitting of spectral lines by a strong magnetic field. He was fired for his efforts, but he was later vindicated: he won the 1902 Nobel Prize in Physics for the discovery of what has now become known as the Zeeman effect. As an extension of his thesis research, he began investigating the effect of magnetic fields on a light source. He discovered that a spectral line is split into several components in the presence of a magnetic field. Lorentz first heard about Zeeman's observations on Saturday October 1896 at the meeting of the Royal Netherlands Academy of Arts and Sciences in Amsterdam, where these results were communicated by Kamerlingh Onnes. The next Monday, Lorentz called Zeeman into his office and presented him with an explanation of his observations, based on Lorentz's theory of electromagnetic radiation. The importance of Zeeman's discovery soon became apparent. It confirmed Lorentz's prediction about the polarization of light emitted in the presence of a magnetic field. Thanks to Zeeman's work it became clear that the oscillating particles that according to Lorentz were the source of light emission were negatively charged, and were a thousandfold lighter than the hydrogen atom. This conclusion was reached well before Thomson's discovery of the electron. The Zeeman effect thus became an important tool for elucidating the structure of the atom. - Wikipedia By Unknown - manehrenfest.jpg (inactive), Copyrighted free use,

16 Zeeman Effect This energy splitting is a linear function of the strength of the magnetic field, and depends on the value of m l. 4 1 me e E = n, m m l l B 2n + 2m ( 4 ) 0

17 Selection Rules For transitions between energy states, angular momentum must be conserved. The photon has an angular momentum of ħ, so the transitions must only be between states with ldifference of 1, which then limits difference of the z-projection of angular momentum also to 1 or less. = 1 m = 0, 1