EE201/MSE207 Lecture 9 Quantum mechanics in three dimensions - - PowerPoint PPT Presentation

EE201/MSE207 Lecture 9 Quantum mechanics in three dimensions

Schrödinger equation: (still only one particle) Natural generalization

𝑗ℏ 𝜖Ψ(𝑦, 𝑧, 𝑨, 𝑢) 𝜖𝑢 = 𝐼 Ψ 𝐼 = 𝑞2 2𝑛 + 𝑊 𝑦, 𝑧, 𝑨, 𝑢 = 𝑞𝑦

2 +

𝑞𝑧

2 +

𝑞𝑨

2

2𝑛 + 𝑊 𝑦, 𝑧, 𝑨, 𝑢

Hamiltonian Momentum

𝑞𝑦 = −𝑗ℏ 𝜖 𝜖𝑦 , 𝑞𝑧 = −𝑗ℏ 𝜖 𝜖𝑧 , 𝑞𝑨 = −𝑗ℏ 𝜖 𝜖𝑨 𝑞 = −𝑗ℏ𝛼

(nabla or del) (sometimes −𝑗ℏ𝛼) SE

𝑗ℏ 𝜖Ψ 𝜖𝑢 = − ℏ2 2𝑛 𝛼2Ψ + 𝑊 Ψ

𝛼2Ψ = 𝜖2Ψ 𝜖𝑦2 + 𝜖2Ψ 𝜖𝑧2 + 𝜖2Ψ 𝜖𝑨2 = ΔΨ

(Laplacian)

usually no t-dependence

If 𝑊(𝑦, 𝑧, 𝑨) (potential energy does not depend on time), then simplification

TISE and general solution of SE

TISE

− ℏ2 2𝑛 𝛼2𝜔𝑜 𝑠 + 𝑊 𝜔𝑜 𝑠 = 𝐹𝑜𝜔𝑜 𝑠

General solution of SE

Ψ 𝑠, 𝑢 =

𝑜

𝑑𝑜 𝜔𝑜 𝑠 exp −𝑗 𝐹𝑜 ℏ 𝑢

𝑠 = (𝑦, 𝑧, 𝑨)

𝐼𝜔𝑜 = 𝐹𝑜𝜔𝑜

Normalization

−∞ ∞

Ψ 2 𝑒𝑦 𝑒𝑧 𝑒𝑨 = 1

−∞ ∞

𝜔𝑜 2 𝑒𝑦 𝑒𝑧 𝑒𝑨 = 1

𝑜

𝑑𝑜 2 = 1

Separation of variables in Cartesian coordinates

Simplification if 𝑊

𝑠 = 𝑊

1 𝑦 + 𝑊 2 𝑧 + 𝑊 3 𝑨 ;

then 3D TISE can be replaced with three 1D equations

− ℏ2 2𝑛 𝜖2𝜔 𝜖𝑦2 + 𝜖2𝜔 𝜖𝑧2 + 𝜖2𝜔 𝜖𝑨2 + 𝑊 𝑠 𝜔 = 𝐹 𝜔

TISE Look for (assume)

𝜔 𝑠 = 𝜔1 𝑦 𝜔2 𝑧 𝜔3 𝑨

Divide TISE by 𝜔, then

− ℏ2 2𝑛 𝜖2𝜔1 𝑦 𝜖𝑦2 𝜔1 𝑦 + 𝜖2𝜔2 𝑧 𝜖𝑧2 𝜔2 𝑧 + 𝜖2𝜔3 𝑨 𝜖𝑨2 𝜔3 𝑨 + 𝑊

1 𝑦 + 𝑊 2 𝑧 + 𝑊 3(𝑨) = 𝐹

Then three equations, with 𝐹 = 𝐹1 + 𝐹2 + 𝐹3

− ℏ2 2𝑛 𝜖2𝜔1 𝑦 𝜖𝑦2 𝜔1 𝑦 + 𝑊

1 𝑦 = 𝐹1

and two similar equations for 𝑧 and 𝑨 (not in the textbook)

Simplification if 𝑊 𝑠 = 𝑊

1 𝑦 + 𝑊 2 𝑧 + 𝑊 3 𝑨

(cont.)

Rewrite as usual

− ℏ2 2𝑛 𝜖2𝜔1 𝑦 𝜖𝑦2 + 𝑊

1 𝑦 𝜔1 𝑦 = 𝐹1 𝜔1 𝑦

− ℏ2 2𝑛 𝜖2𝜔2 𝑧 𝜖𝑧2 + 𝑊

2 𝑧 𝜔2 𝑧 = 𝐹2 𝜔2 𝑧

− ℏ2 2𝑛 𝜖2𝜔3 𝑨 𝜖𝑨2 + 𝑊

3 𝑨 𝜔3 𝑨 = 𝐹3 𝜔3 𝑨

𝐹 = 𝐹1 + 𝐹2 + 𝐹3 𝜔 𝑠 = 𝜔1 𝑦 𝜔2 𝑧 𝜔3(𝑨)

Each equation has many solutions

𝜔𝑙,𝑚,𝑛 𝑠 = 𝜔1,𝑙 𝑦 𝜔2,𝑚 𝑧 𝜔3,𝑛(𝑨)

Energy

𝐹 = 𝐹𝑦,𝑙 + 𝐹𝑧,𝑚 + 𝐹𝑨,𝑛

(replaced 1,2,3 with 𝑦, 𝑧, 𝑨) General solution

Ψ 𝑠, 𝑢 =

𝑙, 𝑚, 𝑛

𝑑𝑙,𝑚,𝑛 𝜔1,𝑙 𝑦 𝜔2,𝑚 𝑧 𝜔3,𝑛(𝑨) exp −𝑗 𝐹𝑦,𝑙 + 𝐹𝑧,𝑚 + 𝐹𝑨,𝑛 ℏ 𝑢

Examples

Unfortunately, not many examples when this trick is useful Semiconductor quantum well, quantum wire, quantum dot

(terminology for semiconductor structures is slightly different than in QM)

quantum well (QW), 2D electron gas (2DEG) electrons do not move in z-direction, free motion in x and y z x quantum wire (QWi), 1D electrons electrons move only in x-direction, restricted along y and z quantum dot (QD), 0D electrons motion is restricted in all direction (x, y, and z) Only the first case (QW) can be truly represented as 𝑊

1 𝑦 + 𝑊 2 𝑧 + 𝑊 3 𝑨 ;

however, other cases can also be treated in this way approximately

Semiconductor Quantum Well

z 𝑊 𝑠 = 𝑊 𝑨 = 0 + 0 + 𝑊

3 𝑨

z

(finite depth QW along z)

Wavefunctions

𝜔(𝑦, 𝑧, 𝑨) = 𝜔𝑜 𝑨 𝑓𝑗𝑙𝑦𝑦𝑓𝑗𝑙𝑧𝑧 1

2𝜌

• r 1

2𝜌ℏ

𝐹 = 𝐹𝑜 + ℏ2𝑙𝑦

2

2𝑛 + ℏ2𝑙𝑧

2

2𝑛

If infinite depth, a 𝑊 𝑠 = 0, 0 ≤ 𝑨 ≤ 𝑏 ∞,

• therwise

then

𝜔(𝑦, 𝑧, 𝑨) =

2 𝑏 sin 𝑜𝜌 𝑏 𝑨

𝑓𝑗𝑙𝑦𝑦𝑓𝑗𝑙𝑧𝑧 1

2𝜌

• r 1

2𝜌ℏ

𝐹 = 𝑜2𝜌2ℏ2 2𝑛𝑏2 + ℏ2𝑙𝑦

2

2𝑛 + ℏ2𝑙𝑧

2

2𝑛

𝑏

Rectangular Quantum Wire

x If finite depth in y and z directions, then we cannot use this trick. However, it works for infinite depth. Assume 𝑊 𝑦, 𝑧, 𝑨 = 0, if 0 ≤ 𝑨 ≤ 𝑏 0 ≤ 𝑧 ≤ 𝑐 ∞,

• therwise

𝜔 𝑦, 𝑧, 𝑨 = 2 𝑏 2 𝑐 sin 𝑜𝑨𝜌 𝑏 𝑨 sin 𝑜𝑧𝜌 𝑐 𝑧 𝑓𝑗𝑙𝑦𝑦 1 2𝜌

• r

1 2𝜌ℏ

If not rectangular and/or finite depth, then still 2+1 dimensions

𝐹 = 𝑜𝑨

2 𝜌2ℏ2

2𝑛𝑏2 + 𝑜𝑧

2 𝜌2ℏ2

2𝑛𝑐2 + ℏ2𝑙𝑦

2

2𝑛

𝑏 𝑐

Rectangular (cuboid) Quantum Dot

Again need to assume infinite depth 𝑊 𝑦, 𝑧, 𝑨 = 0, if 0 ≤ 𝑨 ≤ 𝑏 0 ≤ 𝑧 ≤ 𝑐 0 ≤ 𝑦 ≤ 𝑑 ∞,

• therwise

𝜔 𝑦, 𝑧, 𝑨 = 2 𝑏 2 𝑐 2 𝑑 sin 𝑜𝑨𝜌 𝑏 𝑨 sin 𝑜𝑧𝜌 𝑐 𝑧 sin 𝑜𝑦𝜌 𝑑 𝑦 𝐹 = 𝑜𝑨

2

𝑏2 + 𝑜𝑧

2

𝑐2 + 𝑜𝑦

2

𝑑2 𝜌2ℏ2 2𝑛

Degeneracy if 𝑏, 𝑐, and 𝑑 are equal or commensurate. In semiconductors 𝑛 is effective mass. 𝑏 𝑑 𝑐

Another example: 3D oscillator (e.g., atom in a lattice)

𝑊 𝑠 = 1 2 𝑛𝜕𝑦

2𝑦2 + 1

2 𝑛𝜕𝑧

2𝑧2 + 1

2 𝑛𝜕𝑨

2𝑨2

𝐹𝑜𝑦,𝑜𝑧,𝑜𝑨 = 𝑜𝑦 +

1 2 ℏ𝜕𝑦 + 𝑜𝑧 + 1 2 ℏ𝜕𝑧+ 𝑜𝑨 + 1

2 ℏ𝜕𝑨

Again, degeneracy if 𝜕𝑦, 𝜕𝑧, or 𝜕𝑨 are equal or commensurate.

Spherically symmetric potential (similar trick)

𝑊 𝑠 = 𝑊( 𝑠 )

Most important for atoms Then it is natural to look for

𝜔 𝑠, 𝜄, 𝜒 = 𝑆 𝑠 𝑍(𝜄, 𝜒)

where 𝑠, 𝜄, 𝜒 are spherical coordinates TISE − ℏ2 2𝑛 𝛼2𝜔 + 𝑊 𝑠 𝜔 = 𝐹 𝜔 Rewriting Laplacian in spherical coordinates − ℏ2 2𝑛 1 𝑠2 𝜖 𝜖𝑠 𝑠2 𝜖𝜔 𝜖𝑠 + 1 𝑠2 sin 𝜄 𝜖 𝜖𝜄 sin 𝜄 𝜖𝜔 𝜖𝜄 + 1 𝑠2 sin2 𝜄 𝜖2𝜔 𝜖𝜒2 + 𝑊 𝑠 𝜔 = 𝐹 𝜔 Divide by 𝜔 = 𝑆𝑍 and multiply by −2𝑛𝑠2/ℏ2 1 𝑆 𝜖 𝜖𝑠 𝑠2 𝜖𝑆 𝜖𝑠 − 2𝑛𝑠2 ℏ2 𝑊 𝑠 − 𝐹 + 1 𝑍 1 sin 𝜄 𝜖 𝜖𝜄 sin 𝜄 𝜖𝑍 𝜖𝜄 + 1 sin2 𝜄 𝜖2𝑍 𝜖𝜒2 = 0

𝑚 (𝑚 + 1) −𝑚 (𝑚 + 1)

(so far just a notation) const const combine

Assume 𝑍 𝜄, 𝜒 = Θ 𝜄 Φ(𝜒), again separation of variables 1 Θ(𝜄) 1 sin 𝜄 𝜖 𝜖𝜄 sin 𝜄 𝜖Θ 𝜖𝜄 + 𝑚 𝑚 + 1 sin2 𝜄 + 1 Φ(𝜒) 𝜖2Φ 𝜖𝜒2 = 0

−𝑛2 𝑛2

const const

Φ 𝜒 = 𝑓𝑗𝑛𝜒, 𝑛 = 0, ±1, ±2, …

(since should be periodic with 2𝜌) This is why 𝑛 is integer.

Θ 𝜄 = 𝐵 𝑄

𝑚 𝑛(cos 𝜄)

Associated Legendre function This is why 𝑚 is integer. 𝑚 = 0, 1, 2, … (integer) 𝑛 = −𝑚, −𝑚 + 1, … 0, … 𝑚 − 1, 𝑚

𝑚: anguar momentum quantum number

(azimuthal q. n. , orbital q. n. )

𝑛: magnetic quantum number 𝑍

𝑚 𝑛 𝜄, 𝜒 = Θ𝑚 𝑛 𝜄 Φ𝑛(𝜒)

are called spherical harmonics

These function are the same for any spherically symmetric potential 𝑊(𝑠).

Spherical harmonics 𝑍

Radial function 𝑆

𝜔 = 𝑆 𝑠 𝑍(𝜄, 𝜒)

Let us introduce 𝑣 𝑠 = 𝑠 𝑆 𝑠 , for this function the equation is

− ℏ2 2𝑛 𝑒2𝑣 𝑒𝑠2 + 𝑊 𝑠 + ℏ2 2𝑛 𝑚 𝑚 + 1 𝑠2 𝑣 = 𝐹 𝑣

centrifugal term So, the equation for 𝑣(𝑠) is similar to 1D TISE, but with the centrifugal term. It has some solutions, depending on 𝑚 (orbital q.n.) and 𝑜 (solution index). Corresponding energy: 𝐹𝑜,𝑚 . Overall, 3 quantum numbers: 𝑜, 𝑚, 𝑛. However, energy depends only on 𝑜 and 𝑚.

Hydrogen atom

𝑊 𝑠 = − 𝑓2 4𝜌𝜁0 1 𝑠

Consider only bound states (since atom)  𝐹 < 0 Effective potential

− 𝑓2 4𝜌𝜁0 1 𝑠 + ℏ2 2𝑛 𝑚 𝑚 + 1 𝑠2

Accidentally, for this potential 𝐹𝑜,𝑚 is highly degenerate

𝐹𝑜 = − 𝑛 2ℏ2 𝑓2 4𝜌𝜁0

2

1 𝑜2 = 𝐹1 𝑜2 𝑜 = 1, 2, 3, … 𝐹1 = −13.6 eV (ground state)

𝑜 = 1, 2, 3, … 𝑚 = 0, 1, … 𝑜 − 1 𝑛 = 0, ±1, ±2, … ± 𝑚

Total degeneracy: 𝑚=0

𝑜−1(2𝑚 + 1) = 𝑜2

Ground state: 𝜔100 𝑠, 𝜄, 𝜒 =

1 𝜌𝑏3 𝑓−𝑠/𝑏 𝑏 = 4𝜌𝜁0ℏ2 𝑛𝑓2 = 0.53 Å

Bohr radius

(almost same theory for dopant levels and excitons) principal azimuthal (ang.mom.) magnetic

𝑊(𝑠)

Hydrogen atom (cont.)

𝜔𝑜𝑚𝑛 𝑠, 𝜄, 𝜒 = 1 𝑠 𝜍𝑚+1𝑓−𝜍𝑤 𝜍 × 𝑍

𝑚 𝑛(𝜄, 𝜒)

𝜍 = 𝑠 𝑏𝑜

𝑏 = 4𝜌𝜁0ℏ2 𝑛𝑓2 (Bohr radius, 0.53 ∙ 10−10 m) 𝑤 𝜍 is some polynomial of degree 𝑜 − 𝑚 − 1 (related to generalized Laguerre polynomial)

Spectrum

ℏ𝜕ph = 𝐹𝑗 − 𝐹

𝑔 = −13.6 eV

1 𝑜𝑗

2 − 1

𝑜𝑔

2

𝐹𝑜 = − 𝑛 2ℏ2 𝑓2 4𝜌𝜁0

2

1 𝑜2

4

…

1 2 3 5 Balmer series (visible, 1885) Lyman series (ultraviolet, 1906-14) Paschen series (infrared, 1908)

1 𝜇 = 𝑆 1 𝑜𝑔

2 − 1

𝑜𝑗

2

Rydberg formula, 1888 Rydberg constant 𝑆 = 𝑛 4𝜌𝑑ℏ3 𝑓2 4𝜌𝜁0

2

1.1 ∙ 107m−1