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6.8: Schrödinger Equation in Spherical Coordinates

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Another important eigenvalue problem in physics is the Schrödinger equation. The time-dependent Schrödinger equation is given by

iΨt=22m2Ψ+VΨ.

Here Ψ(r,t) is the wave function, which determines the quantum state of a particle of mass m subject to a (time independent) potential, V(r). From Planck’s constant, h, one defines =h2π. The probability of finding the particle in an infinitesimal volume, dV, is given by |Ψ(r,t)|2dV, assuming the wave function is normalized,

all space |Ψ(r,t)|2dV=1

One can separate out the time dependence by assuming a special form, Ψ(r,t)=ψ(r)eiEt/, where E is the energy of the particular stationary state solution, or product solution. Inserting this form into the time-dependent equation, one finds that ψ(r) satisfies the time-independent Schrödinger equation,

22m2ψ+Vψ=Eψ

Assuming that the potential depends only on the distance from the origin, V=V(ρ), we can further separate out the radial part of this solution using spherical coordinates. Recall that the Laplacian in spherical coordinates is given by

2=1ρ2ρ(ρ2ρ)+1ρ2sinθθ(sinθθ)+1ρ2sin2θ2ϕ2.

Then, the time-independent Schrödinger equation can be written as

22m[1ρ2ρ(ρ2ψρ)+1ρ2sinθθ(sinθψθ)+1ρ2sin2θ2ψϕ2]=[EV(ρ)]ψ.

Let’s continue with the separation of variables. Assuming that the wave function takes the form ψ(ρ,θ,ϕ)=R(ρ)Y(θ,ϕ), we obtain

22m[Yρ2ddρ(ρ2dRdρ)+Rρ2sinθθ(sinθYθ)+Rρ2sin2θ2Yϕ2]=RY[EV(ρ)]ψ.

Dividing by ψ=RY, multiplying by 2mρ22, and rearranging, we have

1Rddρ(ρ2dRdρ)2mρ22[V(ρ)E]=1YLY,

where

L=1sinθθ(sinθθ)+1sin2θ2ϕ2.

We have a function of ρ equal to a function of the angular variables. So, we set each side equal to a constant. We will judiciously write the separation constant as (+1). The resulting equations are then

ddρ(ρ2dRdρ)2mρ22[V(ρ)E]R=(+1)R,1sinθθ(sinθYθ)+1sin2θ2Yϕ2=(+1)Y.

The second of these equations should look familiar from the last section. This is the equation for spherical harmonics,

Ym(θ,ϕ)=2+12(m)!(+m)!Pmeimϕ.

So, any further analysis of the problem depends upon the choice of potential, V(ρ), and the solution of the radial equation. For this, we turn to the determination of the wave function for an electron in orbit about a proton.

Example 6.8.1: The Hydrogen Atom - =0 States

Historically, the first test of the Schrödinger equation was the determination of the energy levels in a hydrogen atom. This is modeled by an electron orbiting a proton. The potential energy is provided by the Coulomb potential,

V(ρ)=e24πϵ0ρ.

Thus, the radial equation becomes

ddρ(ρ2dRdρ)+2mρ22[e24πϵ0ρ+E]R=(+1)R.

Solution

Before looking for solutions, we need to simplify the equation by absorbing some of the constants. One way to do this is to make an appropriate change of variables. Let ρ= ar. Then, by the Chain Rule we have

ddρ=drdρddr=1addr.

Under this transformation, the radial equation becomes

ddr(r2dudr)+2ma2r22[e24πϵ0ar+E]u=(+1)u,

where u(r)=R(ρ). Expanding the second term,

2ma2r22[e24πϵ0ar+E]u=[mae22πϵ02r+2mEa22r2]u,

we see that we can define

a=2πϵ02me2ϵ=2mEa22=2(2πϵ0)22me4E.

Using these constants, the radial equation becomes

ddr(r2dudr)+ru(+1)u=ϵr2u.

Expanding the derivative and dividing by r2,

u+2ru+1ru(+1)r2u=ϵu

The first two terms in this differential equation came from the Laplacian. The third term came from the Coulomb potential. The fourth term can be thought to contribute to the potential and is attributed to angular momentum. Thus, is called the angular momentum quantum number. This is an eigenvalue problem for the radial eigenfunctions u(r) and energy eigenvalues ϵ.

The solutions of this equation are determined in a quantum mechanics course. In order to get a feeling for the solutions, we will consider the zero angular momentum case, =0 :

u+2ru+1ru=ϵu

Even this equation is one we have not encountered in this book. Let’s see if we can find some of the solutions.

First, we consider the behavior of the solutions for large r. For large r the second and third terms on the left hand side of the equation are negligible. So, we have the approximate equation

uϵu=0.

Therefore, the solutions behave like u(r)=e±ϵ for large r. For bounded solutions, we choose the decaying solution.

This suggests that solutions take the form u(r)=v(r)eer for some unknown function, v(r). Inserting this guess into Equation (???), gives an equation for v(r) :

rv+2(1ϵr)v+(12ϵ)v=0.

Next we seek a series solution to this equation. Let

v(r)=k=0ckrk.

Inserting this series into Equation (???), we have

k=1[k(k1)+2k]ckrk1+k=1[12ϵ(k+1)]ckrk=0.

We can re-index the dummy variable in each sum. Let k=m in the first sum and k=m1 in the second sum. We then find that

k=1[m(m+1)cm+(12mϵ)cm1]rm1=0.

Since this has to hold for all m1,

cm=2mϵ1m(m+1)cm1.

Further analysis indicates that the resulting series leads to unbounded solutions unless the series terminates. This is only possible if the numerator, 2mϵ1, vanishes for m=n,n=1,2. Thus,

ϵ=14n2.

Since ϵ is related to the energy eigenvalue, E, we have

En=me42(4πϵ0)22n2.

Inserting the values for the constants, this gives

En=13.6eVn2.

This is the well known set of energy levels for the hydrogen atom.

The corresponding eigenfunctions are polynomials, since the infinite series was forced to terminate. We could obtain these polynomials by iterating the recursion equation for the cm ’s. However, we will instead rewrite the radial equation (???).

Let x=2ϵr and define y(x)=v(r). Then

ddr=2ϵddx.

This gives

2ϵxy+(2x)2ϵy+(12ϵ)y=0.

Rearranging, we have

xy+(2x)y+12ϵ(12ϵ)y=0.

Noting that 2ϵ=1n, this equation becomes

xy+(2x)y+(n1)y=0.

The resulting equation is well known. It takes the form

xy+(α+1x)y+ny=0.

Solutions of this equation are the associated Laguerre polynomials. The solutions are denoted by Lαn(x). They can be defined in terms of the Laguerre polynomials,

Ln(x)=ex(ddx)n(exxn).

The associated Laguerre polynomials are defined as

Lmnm(x)=(1)m(ddx)mLn(x).

Note

The Laguerre polynomials were first encountered in Problem 2 in Chapter 5 as an example of a classical orthogonal polynomial defined on [0,) with weight w(x)=ex. Some of these polynomials are listed in Table 6.8.1 and several Laguerre polynomials are shown in Figure 6.8.1.

Note

The associated Laguerre polynomials are named after the French mathematician Edmond Laguerre (1834-1886). In most derivation in quantum mechanicsa=a02. where a0=4πϵ0h2me2 is the Bohr radius and a0=5.2917×1011 m.

Comparing Equation (???) with Equation (???), we find that y(x)=L1n1(x). In summary, we have made the following transformations:

  1. R(ρ)=u(r),ρ=ar.
  2. u(r)=v(r)eϵr.
  3. v(r)=y(x)=L1n1(x),x=2ϵr.
Table 6.8.1: Associated Laguerre Functions, Lmn(x). Note that L0n(x)=Ln(x).
Lmn(x)
L00(x) 1
L01(x) 1x
L02(x) 12(x24x+2)
L03(x) 16(x3+9x218x+6)
L10(x) 1
L11(x) 2x
L12(x) 12(x26x+6)
L13(x) 16(x3+3x236x+24)
L20(x) 1
L21(x) 3x
L22(x) 12(x28x+12)
L23(x) 112(2x3+30x2120x+120)
clipboard_e4143a4770661f00320e1331fd9aaeee5.png
Figure 6.8.1: Plots of the first few Laguerre polynomials.

Therefore,

R(ρ)=eϵρ/aL1n1(2ϵρ/a).

However, we also found that 2ϵ=1/n. So,

R(ρ)=eρ/2naL1n1(ρ/na).

In Figure 6.8.2 we show a few of these solutions.

clipboard_ee5adede8d78d06b6d09221d6e95fbdf2.png
Figure 6.8.2: Plots of R(ρ) for a=1 and n=1,2,3,4 for the =0 states.

Example 6.8.2

Find the 0 solutions of the radial equation.

Solution

For the general case, for all 0, we need to solve the differential equation

u+2ru+1ru(+1)r2u=ϵu

Instead of letting u(r)=v(r)eϵr, we let

u(r)=v(r)reϵr.

This lead to the differential equation

rv+2(+1ϵr)v+(12(+1)ϵ)v=0.

as before, we let x=2ϵ r to obtain

xy+2[+1x2]v+[12ϵ(+1)]v=0.

Noting that 2ϵ=1/n, we have

xy+2[2(+1)x]v+(n(+1))v=0

We see that this is once again in the form of the associate Laguerre equation and the solutions are

y(x)=L2+1n1(x).

So, the solution to the radial equation for the hydrogen atom is given by

R(ρ)=reϵrL2+1n1(2ϵr)=(ρ2na)eρ/2naL2+1n1(ρna).

Interpretations of these solutions will be left for your quantum mechanics course.


This page titled 6.8: Schrödinger Equation in Spherical Coordinates is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Russell Herman via source content that was edited to the style and standards of the LibreTexts platform.

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