Important Points From Practice

Question 1

Best way to find (ψ,(p ̂²/2M) ψ) for ψs with an awkward derivative?

Answer 1

(ψ,(p ̂²/2M) ψ)=(1/2M) (p ̂ψ,p ̂ψ)

Question 2

How do you find the following integral:

∫_(-∞)^∞(x^m/(1+(x/b)² )ⁿ)dx

When m is even

Answer 2

1. sub u=(x/b)
2. sub u=tanθ, du=(1/cos²⁡θ)dθ and change integration bounds to -π/2<θ<π/2
3. use the formula:
   2∫₀^(π/2)(sinθ)^(2p-1) (cosθ)^(2q-1)dθ=Γ(p)Γ(q)/(Γ(p+q))

Question 3

A quick way to find the following integral:

∫ψ(x)(dψ(x)/dx)dx

Answer 3

∫ψ(x)(dψ(x)/dx)dx=1/2∫d/dx (ψ² (x))dx=ψ²/2+C

Question 4

Harmonic expansions of potential are usually done around the ________ point of the potential.

Answer 4

minimum

Question 5

When finding bound states in a semi-infinite potential well, it is usually convenient to substitute ______ as u and ______ as v when considering boundary condition constraints.

Answer 5

kL=u
κL=v
where
k=(2ME)^1/2/ℏ
wavenumber inside well (0 < x < L)
κ=(2M(V₀-E))^1/2/ℏ
wavenumber in semi-infinite barrier (X>L)

Question 6

When finding bound states in a semi-infinite potential well, it is also convenient to subsitute:
u²+v²=_____ where b=_____

Answer 6

u²+v²=1/b²
where
b=ℏ/(L(2MV₀)^(1/2) )

Question 7

How to find commutative properties of a commutator [A ̂,B ̂]?

Answer 7

Apply it to a general function, as with:
[x ̂,p ̂_x ]f(x)=x ̂p ̂_x f(x)-p ̂_x (x ̂f(x))

p ̂_x in the second term on the right had side also operates on x!

Question 8

[m ̂,n ̂]=?

m,n∈{x ̂,y ̂,z ̂ }, position operators.

Answer 8

[m ̂,n ̂]=0

Position operators commute because they operate as multiplication by a scalar (x, y, z) of the function.

Question 9

[m ̂,p ̂_n]=?

m,n∈{x ̂,y ̂,z ̂ }, position operators.

Answer 9

[m ̂,p ̂_n]=iℏδmn

As in [x ̂,p ̂_x ]=iℏ, because p ̂_x operates nontrivially on x ̂.

Question 10

[p ̂_m,p ̂_n]=?

m,n∈{x ,y, z}

Answer 10

[p ̂_m,p ̂_n]=0

This happens because derivatives of independent coordinates are commutative (e.g. d/dx d/dy=d/dy d/dx and p ̂_m=-iℏ(d/dm).

Question 11

[L ̂_m,L ̂_n ]=?

Answer 11

[L ̂_m,L ̂_n ]=iℏL ̂_l

Right had side obeys right hand rule.

e.g. [L ̂_x,L ̂_y ]=iℏL ̂_z

Question 12

When m≠n:

[L ̂_m,p ̂_n ]=?

Answer 12

[L ̂_m,p ̂_n ]=iℏp ̂_l

Right had side obeys right hand rule.

e.g. [L ̂_x,p ̂_y ]=iℏp ̂_z

Question 13

When m≠n:

[L ̂_m,n ̂ ]=?

m,n∈{x ̂,y ̂,z ̂ }

Answer 13

[L ̂_m,n ̂]=iℏl ̂_l

Right had side obeys right hand rule.
l∈{x ̂,y ̂,z ̂ }

e.g. [L ̂_x,y ̂]=iℏz ̂

Question 14

When m=n:

[L ̂_m,p ̂_m ]=?

m,n∈{x ̂,y ̂,z ̂ }

Answer 14

[L ̂_m,p ̂_m ]=0

Question 15

When m=n:

[L ̂_m,m ̂ ]=?

m,n∈{x ̂,y ̂,z ̂ }

Answer 15

[L ̂_m,m ̂ ]=0

Question 16

Given the hamiltonian operator:
H ̂=p ̂²/2M+V(x,y,z)
what are the conditions for [H ̂,L ̂ ]=0?

Answer 16

[p ̂²/2M, L ̂]=0 unconditionally.

[V ̂,L ̂ ]=0 when ∇V ̂∥e ̂_r. This happens when V ̂=V(r) ( V ̂ is a central potential).

Question 17

Do eigenstates of a H ̂ which satisfies:
H ̂(r ̂₁,r ̂₂,p ̂₁,p ̂₂)=H ̂(r ̂₂,r ̂₁,p ̂₂,p ̂₁)

satisfy:
ψ(r ̂₁,r ̂₂ )=ψ((r₂ ) ̂,r ̂₁)?

This is a H ̂ which describes a system of two identical particles, such as electrons in a He atom.

Answer 17

Not necessarily!

A good counterexample are eigenstates of a system of two identical particels in a 1D harmonic oscillator. These igenstates are not necessarily eigenstates of the permutation operator p ̂₁₂.

Question 18

Given that a H ̂ describing a system of 2 identical particles satisfies:

[H ̂,p ̂₁₂ ]=0

Are all eigenstates of H ̂ symmetric\antisymmetric to permutation?

Answer 18

No! The system described by H ̂ can have degenerated eigenvalues.

This means that not all eigenfunctions of H ̂ are also eigenfunctions of p ̂₁₂.

A good counterexample are eigenstates of a H ̂ of two identical particles in a harmonic oscillator.

Question 19

What is a unitary operator?

Answer 19

A unitary operator U ̂ is an operator which satisfies, for every ψ₁, ψ₂:

(U ̂ψ₁,U ̂ψ₂ )=(ψ₁,ψ₂ )

An operator which conserves value of internal product. e.g. e^iωt.

Question 20

The hermitian conjugate of a unitary operator U satisfies…

Answer 20

U ̂=U^(-1)

Question 21

The absolute value |λ| of all eigenvalues λ of a unitary operator satisfy:

Answer 21

|λ|=1

Question 22

Condition for orthonormality between Y_lm and Y_l’m’?

Answer 22

(Y_lm,Y_{(l^’ m)} )=δ_{(ll^’ )} δ_(mm^’)

Y_lm are eigenfunctions of L² and L_z which are both hermitian operators. This is why Ylm’s are orthonormal for different m values or l values!

Question 23

Footnote

H ̂ for rotaring rigid body?

Such as a poly atomic molecule, in the non inertial system

Answer 23

H ̂_RB=(L₁²)/2I₁ + (L₂²)/2I₂ + (L₃²)/2I₃

Question 24

H ̂ for symmetrical rotaring rigid body?

Answer 24

H ̂_S=L ̂²/(2I₁ )+1/2 (1/I₃ -1/I₁ ) L ̂₃²

I₁=I₂≠I₃

Question 25

H ̂_RB commutates with \_\_\_\_\_\_\_ and \_\_\_\_\_\_\_, but not with \_\_\_\_\_\_\_.

Answer 25

L ̂², L ̂_z but not with L ̂_i (i=1,2,3...) ## Footnote - [Hrb, Lz] = 0 because each depends on an independent set of coordinates?

Question 26

H ̂_S commutates with \_\_\_\_\_\_\_ and \_\_\_\_\_\_\_, **and also** with \_\_\_\_\_\_\_.

Answer 26

L ̂², L ̂_z and also with L ̂_i (i=1,2,3...)

Question 27

What can be said about the parity of eigenstates of a system described by a H ̂ with an even V(r) (V(**r**)=V(-**r**))?

Answer 27

These eigenstates are either even (ψ(r)=ψ(-r)) or uneven (ψ(-r)=-ψ(r)) about the origin. | They are eigenstates of the parity operator ## Footnote e.g. eignestates of a harmonic oscillator are either even or uneven. parity is determined by the hermit polynomials.