Physics 130b: LectureTime Evolution

§Time Evolution–Observables–The state vector–Spin precession

Spin Angular Momentum

𝑆!| ⟩𝑠 𝑚 = ℏ!𝑠 𝑠 + 1 | ⟩𝑠 𝑚

𝑆"| ⟩𝑠 𝑚 = ℏ𝑚| ⟩𝑠 𝑚

𝑆±| ⟩𝑠 𝑚 = ℏ 𝑠 𝑠 + 1 −𝑚 𝑚 ± 1 | ⟩𝑠 𝑚 ± 1

𝑆± = 𝑆$ ± 𝑖𝑆%

Clicker Question 1

Clicker Question 2

Matrix Representation of Operators

𝐴 ≐ ↑ 𝐴 ↑ ↑ 𝐴 ↓↓ 𝐴 ↑ ↓ 𝐴 ↓

↑ 𝐴 ↑

Student Questions§ I'm not clear on what equation 2.4 is listing, the "explicit labelling of the

rows.”

Let’s find 𝑆-𝑆!| ⟩↑ = ℏ

#| ⟩↑ 𝑆!| ⟩↓ = -

ℏ#| ⟩↓

Let’s find 𝑆-𝑆!| ⟩↑ = ℏ

#| ⟩↑ 𝑆!| ⟩↓ = -

ℏ#| ⟩↓

An operator is always diagonal in it’s own basis. Eigenvectors are unit vectors in their own basis.

Phys3220, U.Colorado at Boulder151

The raising operator operating on the up and down spin states:

0, =­­=¯ ++ SS !

What is the matrix form of the operator S+ ?

A) B) C)÷÷ø

öççè

æ0110

! ÷÷ø

öççè

æ0010

! ÷÷ø

öççè

æ0100

!

D) E) None of these÷÷ø

öççè

æ1110

!

Clicker Question 3

Phys3220, U.Colorado at Boulder149

In spin space, the basis states (eigenstates of S2,

Sz ) are orthogonal:

Are the following matrix elements zero or non-zero?

A) Both are zeroB) Neither are zeroC) The first is zero; second is non-zeroD) The first is non-zero; second is zero

.0=¯­

¯­¯­ zSS 2Clicker

Question 4

Example

𝑆!| ⟩𝑠 𝑚 = ℏ!𝑠 𝑠 + 1 | ⟩𝑠 𝑚

𝑆"| ⟩𝑠 𝑚 = ℏ𝑚| ⟩𝑠 𝑚

Rules of Quantum Mechanics6) The time evolution of a quantum system is determined

by the Hamiltonian of total energy operator 𝐻 𝑡 through the Schrodinger equation

𝑖ℏ !!"| ⟩𝜓 𝑡 = 𝐻 𝑡 | ⟩𝜓 𝑡

SE

𝐻 | ⟩𝐸# = 𝐸# | ⟩𝐸#

𝑖ℏ !!"| ⟩𝜓 𝑡 = 𝐻 | ⟩𝜓 𝑡

| ⟩𝜓 𝑡 =,#

𝑐#𝑒$%&&"/ℏ| ⟩𝐸#

Time-dependent RecipeBelow is the recipe for solving a standard time-

dependent quantum mechanics problem with a time dependent Hamiltonian.

Given1. Diagonalize 𝐻 to determine the energy eigenvectors

and energy eigenvalues2. Write in terms of the energy eigenvectors:3. Multiply each eigenvector by 4. Calculate (prob. of various observables)

)0(y

)0(y!/tiEne-

General Procedure (for time independent potentials)

Given V(x) and Y(x,0)1. Solve TISE and find eigenfunctions yn (x) and

eigenvalues En associated with V(x) 2. Write in Y(x,0) terms of the eigenstates:

3. Multiply each eigenfunction by to get

4. Calculate (prob. of various observables)

!/tiEne-å¥

=

=Y1

)()0,(n

nn xcx y

å¥

=

-=Y1

/)(),(n

tiEnn

nexctx !y

Electron in Uniform B-Field

An electron in a magnetic field 𝐵 = 𝐵!�̂� is initially in a spin state ⟩|χ(0) = 𝑎 ⟩|↑ " + 𝑏 ⟩|↓ ". Which of the following equations correctly

represents the state ⟩|χ(𝑡) of the electron after time t? The Hamiltonian operator is 1𝐻 = −𝛾𝐵! 5𝑆".

A. ⟩|χ(𝑡) = 𝑒#$%!&/( 𝑎 ⟩|↑ " + 𝑏 ⟩|↓ "

B. ⟩|χ(𝑡) = 𝑒)#$%!&/( 𝑎 ⟩|↑ " + 𝑏 ⟩|↓ "

C. ⟩|χ(𝑡) = 𝑒#$%!&/( (𝑎 + 𝑏) ⟩|↑ " + (𝑎 − 𝑏) ⟩|↓ "

D. ⟩|χ(𝑡) = 𝑎𝑒#$%!&/( ⟩|↑ " + 𝑏𝑒)#$%!&/( ⟩|↓ "

E. None of the above

Clicker Question 5: