Physics 401: Quantum Mechanics I

Chapter 3

Are you here today?

A. YesB. NoC. In which basis?

Quantum Mechanics Formalism

Compared to the original ψ(x), the set of numbers c1,c2,c3,…. contains:

A) more information.B) less information.C) the same informationD) cannot be determined/depends.

1

A wavefunction ψ(x) has been expressed as a sum of energy eigenfunctions (un(x)’s):

ψ(x) = cn un(x)

n∑

If we want the inner product of V with itself, <V|V>, to be positive (for nonzero V), what should <A|B> be?

A) A1 B1 +A2 B2 B) A*1 B1 +A*2 B2C)|A1 B1 +A2 B2| D) More than one of these options E) NONE of these makes sense.

Consider a complex vector V:

Where V1 and V2 are complex numbers (they are the “components of V”)€

V ⇔ (V1,V2)

If f(x) and g(x) are wave functions, and c is a constant,

then <cf|g> = ?A) c<f|g>B) c*<f|g>C) |c|<f|g>D) c<f*|g>E) None of these

A vector can be written as a column of its components in a basis ;; likewise a vector in Hilbert space (a wave function) can be written as an infinite column of

its components in a basis of the ψns:

A) B)

C) D) E) zero59

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A =

Ax

Ay

Az

"

#

$ $ $

%

&

' ' ' Ψ =

c1

c2

c3

c4

"

#

$ $ $ $ $ $

%

&

' ' ' ' ' '

The dot product of two vectors A and B is:

The inner product of two wavefunctions,

€

( ˆ x , ˆ y , ˆ z )

€

A ⋅ B = AiBi

i= x,y,z∑

€

dn*cn

n∑

€

dn cnn∑

€

dn2 cn

n∑

2

€

dn2

+ cn2( )

n∑

Ψ = cnψ n

n∑ and Φ = dnψ n ,

n∑ is Φ Ψ = dx Φ*Ψ = ...∫

Particle in an infinite square well potential

Ket Representation Wave Function Representation Matrix Representation

Hamiltonian H

H −

2

2m

d2

dx2

H

E1

0 0

0 E2

0

0 0 E3

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Eigenvalues of

Hamiltonian

Normalized Eigenstates

of Hamiltonian

n ψn

x( ) =2

Lsin

nπL

x⎛⎝⎜

⎞⎠⎟

1

1

0

0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

, 2

0

1

0

0

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

, …

Coefficient of energy

eigenstate

Probability of measuring

P

En

= cn

2

= n ψ2

PE

n

= cn

2

= 2

Lsin

nπL

x

⎛⎝⎜

⎞⎠⎟ψ x( )

0

L

∫ dx

2

PE

n

= cn

2

= 0 1 ( )

c1

cn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

2

Expectation value of

Hamiltonian

True (A) or False (B):The operator i (i.e. multiplying by the constant ) is a Hermitian

operator. i = −1

64

The momentum operator p = i∂∂x

is hermitian,

meaning f | pg = pf | g .

Is p2 hermitian?

A) YesB) No

Properties of Hermitian operators

1. The eigenvalues of Hermitian operators are always real.

2. The expectation values of Hermitian operators are always real.

3. The eigenvectors of Hermitian operators span the Hilbert space.

4. The eigenvectors of Hermitian operators belonging to distinct eigenvalues are orthogonal.

Are you here today?

• The midterm exam will be held in class next Wednesday, February 3

• The exam will use the entire class period, so don’t be late!

• The exam will cover the material in Chapter 3• The exam will be closed-­book, closed notes• All necessary equations will be provided• I will be in Texas, giving a colloquium, so Matt will be proctoring the exam

PHY 402 Midterm

Postulates of Quantum Mechanics(1) The state of a particle is completely represented by a normalized vector in Hilbert space, which we call |ψ>

(2) All physical observables, Q, are associated with Hermitian operators Q, and the expectation value of Q in some state |ψ> is <ψ|Q|ψ>

(3) A measurement of Q on a particle in state |ψ> is certain to return a particular value, λ , iff ("if and only if") Q|ψ> = λ|ψ>, (i.e. if and only if |ψ> is already an eigenvector of Q, with eigenvalue λ)

(3a) If you measure Q in any state |ψ>, you are certain to obtain one of the eigenvalues of Q. The probability of measuring some eigenvalue λ is given by |<uλ|ψ>|2, where |uλ> is defined to be the eigenvector of Q with eigenvalue λ

(3b) After a measurement gives you the value λ, the system will collapse into the state |uλ>

(4) The time evolution of the state |ψ> is given by Schrodinger's equation:

i∂ψ∂t

= H ψ

Suppose |f1> and |f2> are eigenvectors of operator Q, with eigenvalues q1 and q2respectively.Is a|f1>+b|f2> an eigenvector of Q?

A) Yes, alwaysB) No, neverC) Only if a=bD) Only if q1=q2E) Not sure/something else/???

Observable A is measured, and the value a1 is found. What is the system's state immediately after measurement?

A) B) C)D) E)

71

€

Observable A : ˆ A ψ = aψ normalized eigenstates ψ1, ψ2, eigenvalues a1, a2.Observable B: ˆ B φ = bφ normalized eigenstates φ1, φ2, eigenvalues b1, b2.The eigenstates are related by ψ1 = (2φ1 + 3φ2)

13 ψ2 = (3φ1 − 2φ2)

13

€

ψ1

€

ψ2

€

c1ψ1 + c2ψ2 (c1 & c2 non - zero)

€

φ1

€

φ2

Immediately after the measurement of A, the observable B is measured. What is the probability that b1 will be found?

A) 0 B) 1 C) 0.5 D) 2/√13 E) 4/13

72

€

Observable A : ˆ A ψ = aψ normalized eigenstates ψ1, ψ2, eigenvalues a1, a2.Observable B: ˆ B φ = bφ normalized eigenstates φ1, φ2, eigenvalues b1, b2.The eigenstates are related by ψ1 = (2φ1 + 3φ2)

13 ψ2 = (3φ1 − 2φ2)

13

If the grad student failed to measure B, but instead measured A for a second time, what is the probability that the second measurement will yield a1?

A) 0 B) 1 C) 0.5 D) 2/√13 E) 4/1373

€

Observable A : ˆ A ψ = aψ normalized eigenstates ψ1, ψ2, eigenvalues a1, a2.Observable B: ˆ B φ = bφ normalized eigenstates φ1, φ2, eigenvalues b1, b2.The eigenstates are related by ψ1 = (2φ1 + 3φ2)

13 ψ2 = (3φ1 − 2φ2)

13

Δ baryon decay mass spectrum