Help with a question about Bohr and Schrodinger?

Question

both the Bohr and Schrodinger approaches give the same answer for the allowed energy levels of the hydrogen atom. why is Schrodinger approach preferred? give two reasons.

Answer 1

schrodinger gives a great many accurate predictions about a lot of things.  not just the energy levels of hydrogen.  bohr does not.  I cant even remember details of bohr's stuff so i cant really comment, except that not remembering it is an indicator that it's not very spectacular!  schrodinger's work certainly is spectacular!

Answer 2

The problem with Bohr's model is that it can *only* predict the energy levels of hydrogen.

The Schrödinger equation can also explain a multitude of other effects: why multiple electrons stack up in the energy levels the way they do (Pauli exclusion principle), why extra energy levels are observed when atoms are in a magnetic field (Lamb shift), why the atom doesn't collapse in on itself as an orbiting electron radiates away energy as it accelerates around the nucleus (Larmor formula).

As no teacher is going to believe you came up with the above yourself, you're going to have to do some research in order to state this in your own words.

Answer 3

I will start my answer with a brief presentation of the Bohr model then contrast it with a brief presentation of the Schrödinger wave mechanics approach to atomic electrons.

The Bohr Theory of atomic electrons is as follows: -

1. The electron revolves in a circular orbit with the centripetal force supplied by the coulomb interaction between the electron and the nucleus, which may be expressed as: -

Ze².. = mv² = mω²r

___ .... __

4πε0r².. r

where r = radius of allowed orbit

v = electron velocity

ω = corresponding angular velocity

2. The angular momentum A of the electron takes on only values which are integer multiples of ħ, such that: -

A = mvr = mωr² = nħ

3. When an electron makes a transition from one allowed stationary orbit state to another, the Einstein frequency condition hν =E(i) - E(f) is satisfied (i -> initial and f -> final). Thus, the allowed radii are: -

r(n) = 4πε0ħ²n²

........________

....... me²Z

The energy of the electron is partially kinetic and partially potential. If we call the energy zero when the electron is at rest at infinity, its potential energy in the presence of the nucleus is: -

P = - 1 . Ze²

..... __ . ____

.....4πε0 . r

It kinetic energy is: -

K = ½mv² = ½mr²ω²

Since, from Bohr's postulates (above): -

v = nħ/mr

The velocity of the electron in the n'th orbit is: -

v(n) = e²Z

....... ____

....... 2ε0hn

Hence, the kinetic energy is: -

K = .. 1.. Ze²

....... __ .. ___

..... 4πε0 2r

The total energy is E = K + P, and so for the n'th state (after summing the two equations and reducing the expression): -

E(n) = - me⁴Z²

........ ________

........ 8ε0²h²nZ²

This calculation needs to be corrected for reduced mass of an orbital system. Thus, for a nucleus of mass ‘M’ and electron of mass ‘m’, the reduced mass is given by: -

m(r) = m/(1 + m/M)

Hence: -

E(n) = - m(r)e⁴Z²

........ ________

........ 8ε0²h²nZ²

The Bohr model of atomic electrons was published in 1913; it was a classical model with a quantum restriction on energy level shifts or jumps. This model could only give limited 'correct' answers for the one electron hydrogen atom.

Over the next decade, it was discovered that the subatomic world's particles exhibit a wave and particle duality. Thus, in 1924 de Broglie, in his doctoral thesis: 'Recherches sur la théorie des quanta', proposed that particles have a wavelength dependent upon their momentum 'p', given by: -

λ = h/p

Three years later, electron diffraction was confirmed at 'Bell Labs' when Clinton Joseph Davisson and Lester Halbert Germer guided their electron beam through a crystalline grid.

in 1926, the Austrian mathematician  Erwin Schrödinger discovered an equation, which allowed the successful description of the atomic electrons around a nuclei  of an atom. This equation is: -

iħ.∂Ψ(x, t) = H.Ψ(x, t)

... _______

....∂t

This equation allowed the description of any atomic electron in terms of a wave function or eigen function, which is a probabilistic description of all that is or can be known about the electron. The equation incorporated the potential energy well of the nuclei and the momentum of the electron in its summation. Furthermore, the wave particle duality of sub atomic particles was included within the equation. The solution to the wave function was an expectation value or eigen value for a given observation of the electrons properties. Finally, the equation implicitly includes Heisenberg’s uncertainty principle. Namely, that the spread of momentum ‘p’ and position ‘x’ for a particle are two mutually exclusive variables: -

∆x.∆p ≥ ħ/2

Wikipedia adds, 'In 1926, Erwin Schrödinger, using Louis de Broglie's 1924 proposal that particles behave to an extent like waves, developed a mathematical model of the atom that described the electrons as three-dimensional waveforms, rather than point particles. A consequence of using waveforms to describe electrons is that it is mathematically impossible to obtain precise values for both the position and momentum of a particle at the same time; this became known as the uncertainty principle, formulated by Werner Heisenberg in 1926. In this concept, for each measurement of a position one could only obtain a range of probable values for momentum, and vice versa. Although this model was difficult to visualise, it was able to explain observations of atomic behaviour that previous models could not, such as certain structural and spectral patterns of atoms larger than hydrogen. Thus, the planetary model of the atom was discarded in favor of one that described atomic orbital zones around the nucleus where a given electron is most likely to exist.'

Hence, the Schrödinger wave equation may in principle provide expectation vales for the observation of the properties of all atomic electrons but the Bohr model was restricted to the one electron hydrogen atom. Furthermore, the wave mechanics approach could deal with electron spin and thus provide expectation results for hyperfine spectral observations.