Quantum Mechanics Is Weird

Stern-Gerlach (SG) Experiment (1922)

Ag

Oven heats up silver atoms. 

Some escape.

South

North

Magnetic field

Beam direction

Z-direction

Detector

SG Predictions

A simple model of a silver atom treats it like a tiny magnet.

Heating the silver in the oven will randomize the magnetic orientation of the atom.

The force on an atom is proportional to the z-component of the "spin" of the atom:

F_z \approx \mu_z \frac{\partial B}{\partial z}
Fz≈μz∂B∂zF_z \approx \mu_z \frac{\partial B}{\partial z}

Since the spins are randomized, you'd expect to see a more-or-less uniform splash of silver at your detector.

BUT DO YOU!?

What you predict

What you see

...???!!!???

You see two distinct components!

S_z+
Sz+S_z+
S_z-
Sz−S_z-

Not Weird Enough For Ya?

SGz

beam

S_z+
Sz+S_z+
S_z-
Sz−S_z-

Now let's block one of the streams, and feed it into another SG machine and block an output.

SGz

beam

S_z+
Sz+S_z+

SGz

S_z+
Sz+S_z+

This...makes sense.

A Twist

SGz

beam

S_z+
Sz+S_z+

SGx

Wait what

S_x+
Sx+S_x+
S_x-
Sx−S_x-

And Back Again

SGz

beam

S_z+
Sz+S_z+

SGx

S_x+
Sx+S_x+

SGz

S_z+
Sz+S_z+
S_z-
Sz−S_z-

But...

y-you're dead!

This doesn't happen in regular old boring classical physics.  When you toss a novelty flying disc, you can easily measure the spin in the x- and z-direction.

Not so in quantum mechanics.  Measuring the x-spin destroys information about the z-spin.

And what about the y-direction!?

We might have to use...MATH

Dirac notation! States live in a linear vector space, called a Hilbert space.

Quantum states are

represented by vectors, called kets.

\left|\psi\right\rangle,\left|\phi\right\rangle,\left|\omega\right\rangle...
∣ψ⟩,∣ϕ⟩,∣ω⟩...\left|\psi\right\rangle,\left|\phi\right\rangle,\left|\omega\right\rangle...

Observables, like spin or position or momentum, are represented by operators.

A,B,C...
A,B,C...A,B,C...

Operators act on states.

A\left|\alpha\right\rangle = \left|\beta\right\rangle
A∣α⟩=∣β⟩A\left|\alpha\right\rangle = \left|\beta\right\rangle

States can be added and multiplied by a complex scalar.

c_\alpha \left|\psi\right\rangle + c_\beta \left|\phi\right\rangle = \left|\gamma\right\rangle
cα∣ψ⟩+cβ∣ϕ⟩=∣γ⟩c_\alpha \left|\psi\right\rangle + c_\beta \left|\phi\right\rangle = \left|\gamma\right\rangle

Eigenvalues and Eigenstates

In general, applying an operator to a ket gives you a different ket.

A\left|\alpha\right\rangle = \left|\beta\right\rangle
A∣α⟩=∣β⟩A\left|\alpha\right\rangle = \left|\beta\right\rangle

But sometimes, operating on a ket merely rescales it:

A \left| a \right\rangle = a \left| a \right\rangle
A∣a⟩=a∣a⟩A \left| a \right\rangle = a \left| a \right\rangle

The ket is called an eigenstate, and the scaling factor is the eigenvalue.

Back to Spin

For example:

S_z\left|S_z+\right\rangle = +\frac{\hbar}{2}\left|S_z+\right\rangle
Sz∣Sz+⟩=+ℏ2∣Sz+⟩S_z\left|S_z+\right\rangle = +\frac{\hbar}{2}\left|S_z+\right\rangle
S_z\left|S_z-\right\rangle = -\frac{\hbar}{2}\left|S_z-\right\rangle
Sz∣Sz−⟩=−ℏ2∣Sz−⟩S_z\left|S_z-\right\rangle = -\frac{\hbar}{2}\left|S_z-\right\rangle
S_x\left|S_x+\right\rangle = +\frac{\hbar}{2}\left|S_x+\right\rangle
Sx∣Sx+⟩=+ℏ2∣Sx+⟩S_x\left|S_x+\right\rangle = +\frac{\hbar}{2}\left|S_x+\right\rangle
S_x\left|S_x-\right\rangle = -\frac{\hbar}{2}\left|S_x-\right\rangle
Sx∣Sx−⟩=−ℏ2∣Sx−⟩S_x\left|S_x-\right\rangle = -\frac{\hbar}{2}\left|S_x-\right\rangle

You can uniquely expand an arbitrary ket in terms of eigenkets in the space, which form a basis:

\left|\alpha\right\rangle = \sum\limits_ac_a\left|a\right\rangle
∣α⟩=∑aca∣a⟩\left|\alpha\right\rangle = \sum\limits_ac_a\left|a\right\rangle

Dual Vectors: Bra(c)kets

We can associate each ket with its dual, called a "bra" in the 1920s:

\langle a\left|a\right\rangle
⟨a∣a⟩\langle a\left|a\right\rangle
A\left|a\right\rangle \leftrightarrow \langle a|A^\dagger
A∣a⟩↔⟨a∣A†A\left|a\right\rangle \leftrightarrow \langle a|A^\dagger
c\left|a\right\rangle \leftrightarrow c^*\langle a|
c∣a⟩↔c∗⟨a∣c\left|a\right\rangle \leftrightarrow c^*\langle a|

We create an inner product between a bra and ket, called a...sigh.  A bracket:

The bracket / inner product is a complex number.

\left|a\right\rangle \leftrightarrow \langle a|
∣a⟩↔⟨a∣\left|a\right\rangle \leftrightarrow \langle a|

Bracket Properties

\langle a\left|a\right\rangle \ge 0
⟨a∣a⟩≥0\langle a\left|a\right\rangle \ge 0

(Brackets are positive-definite)

\langle \alpha\left|\beta\right\rangle = \langle \beta\left|\alpha\right\rangle^*
⟨α∣β⟩=⟨β∣α⟩∗\langle \alpha\left|\beta\right\rangle = \langle \beta\left|\alpha\right\rangle^*

(Antisymmetry)

In QM, only the direction, not the scale, matters.  So:

\left|\beta\right\rangle
∣β⟩\left|\beta\right\rangle
c\left|\beta\right\rangle
c∣β⟩c\left|\beta\right\rangle

and

are the same state.

We may as well work with just the normalized state:

\left|\tilde{\beta}\right\rangle = \frac{1}{\sqrt{\langle \beta\left|\beta\right\rangle }} \left|\beta\right\rangle
∣β~⟩=1⟨β∣β⟩∣β⟩\left|\tilde{\beta}\right\rangle = \frac{1}{\sqrt{\langle \beta\left|\beta\right\rangle }} \left|\beta\right\rangle

This is convenient, since now:

\left\langle\tilde{\beta} |\tilde{\beta}\right\rangle = 1
⟨β~∣β~⟩=1\left\langle\tilde{\beta} |\tilde{\beta}\right\rangle = 1

Operator Algebra

A\left|\alpha\right\rangle = B\left|\alpha\right\rangle \forall \alpha \rightarrow A = B
A∣α⟩=B∣α⟩∀α→A=BA\left|\alpha\right\rangle = B\left|\alpha\right\rangle \forall \alpha \rightarrow A = B
(X + Y) + Z = X + (Y + Z)
(X+Y)+Z=X+(Y+Z)(X + Y) + Z = X + (Y + Z)
XY \ne YX
XY≠YXXY \ne YX
X(c_\alpha \left|\psi\right\rangle + c_\beta \left|\phi\right\rangle) = c_\alpha X\left|\psi\right\rangle + c_\beta X \left|\phi\right\rangle
X(cα∣ψ⟩+cβ∣ϕ⟩)=cαX∣ψ⟩+cβX∣ϕ⟩X(c_\alpha \left|\psi\right\rangle + c_\beta \left|\phi\right\rangle) = c_\alpha X\left|\psi\right\rangle + c_\beta X \left|\phi\right\rangle
A^\dagger
A†A^\dagger

is the dual operator and called the Hermitian adjoint of

A
AA
A = A^\dagger
A=A†A = A^\dagger

An operator is Hermitian or self-adjoint if 

(XY)Z = X(YZ)
(XY)Z=X(YZ)(XY)Z = X(YZ)

The Outer Product

We can multiply a ket by a complex number, and we can take the inner product of a bra-ket pair.  We can also form the outer product:

\left|\alpha\right\rangle\langle \beta |
∣α⟩⟨β∣\left|\alpha\right\rangle\langle \beta |

Apply it to a ket:

(\left|\alpha\right\rangle\langle \beta |) \cdot \left|\gamma\right\rangle = \langle \beta \left|\gamma\right\rangle \left|\alpha\right\rangle
(∣α⟩⟨β∣)⋅∣γ⟩=⟨β∣γ⟩∣α⟩(\left|\alpha\right\rangle\langle \beta |) \cdot \left|\gamma\right\rangle = \langle \beta \left|\gamma\right\rangle \left|\alpha\right\rangle

Dirac called this his "associative" axiom.

Brackets and Operators

(\langle \alpha|) \cdot X\left|\beta\right\rangle = (\langle \alpha|X) \cdot \left|\beta\right\rangle = \langle \alpha|X \left|\beta\right\rangle
(⟨α∣)⋅X∣β⟩=(⟨α∣X)⋅∣β⟩=⟨α∣X∣β⟩(\langle \alpha|) \cdot X\left|\beta\right\rangle = (\langle \alpha|X) \cdot \left|\beta\right\rangle = \langle \alpha|X \left|\beta\right\rangle
\langle \alpha|X \left|\beta\right\rangle = \langle \beta|X^\dagger \left|\alpha\right\rangle^*
⟨α∣X∣β⟩=⟨β∣X†∣α⟩∗\langle \alpha|X \left|\beta\right\rangle = \langle \beta|X^\dagger \left|\alpha\right\rangle^*

We can combine operators and kets:

Observables are Hermitian

The eigenvalues of a Hermitian operator are real.  Proof:

A\left|a_1\right\rangle = a_1\left|a_1\right\rangle
A∣a1⟩=a1∣a1⟩A\left|a_1\right\rangle = a_1\left|a_1\right\rangle

But since A is Hermitian,

\langle a_2|A = a_2^*\langle a_2|
⟨a2∣A=a2∗⟨a2∣\langle a_2|A = a_2^*\langle a_2|

Multiply by 

\langle a_2|
⟨a2∣\langle a_2|

Multiply by 

\left| a_1\right\rangle
∣a1⟩\left| a_1\right\rangle

Subtract:

(a_1 - a_2^*)\langle a_2|a_1\rangle = 0
(a1−a2∗)⟨a2∣a1⟩=0(a_1 - a_2^*)\langle a_2|a_1\rangle = 0
\langle a_2| A\left|a_1\right\rangle = \langle a_1| A\left|a_2\right\rangle
⟨a2∣A∣a1⟩=⟨a1∣A∣a2⟩\langle a_2| A\left|a_1\right\rangle = \langle a_1| A\left|a_2\right\rangle

Observables are Hermitian

(a_1 - a_2^*)\langle a_2|a_1\rangle = 0
(a1−a2∗)⟨a2∣a1⟩=0(a_1 - a_2^*)\langle a_2|a_1\rangle = 0
|a_1\rangle = |a_2\rangle:
∣a1⟩=∣a2⟩:|a_1\rangle = |a_2\rangle:
(a_1 - a_1^*) = 0 \rightarrow a_1 = a_1^*
(a1−a1∗)=0→a1=a1∗(a_1 - a_1^*) = 0 \rightarrow a_1 = a_1^*
|a_1\rangle \ne|a_2\rangle:
∣a1⟩≠∣a2⟩:|a_1\rangle \ne|a_2\rangle:
(a_1 - a_2^*)\langle a_2|a_1\rangle = (a_1 - a_2)\langle a_2|a_1\rangle
(a1−a2∗)⟨a2∣a1⟩=(a1−a2)⟨a2∣a1⟩(a_1 - a_2^*)\langle a_2|a_1\rangle = (a_1 - a_2)\langle a_2|a_1\rangle
\rightarrow \langle a_2|a_1\rangle = 0
→⟨a2∣a1⟩=0\rightarrow \langle a_2|a_1\rangle = 0

Thus the eigenvalues of a Hermitian operator are real, and kets with different eigenvalues are orthogonal.

Use Orthonormal Sets

\langle a|b\rangle = \delta_{ab}
⟨a∣b⟩=δab\langle a|b\rangle = \delta_{ab}
\delta_{ab} = \begin{cases} 1, a=b\\ 0, a\ne b \end{cases}
δab={1,a=b0,a≠b\delta_{ab} = \begin{cases} 1, a=b\\ 0, a\ne b \end{cases}

Expand an arbitrary ket into a linear combinations of its eigenkets:

\left|\alpha\right\rangle = \sum\limits_ac_a\left|a\right\rangle
∣α⟩=∑aca∣a⟩\left|\alpha\right\rangle = \sum\limits_ac_a\left|a\right\rangle

Act on the left with an eigenvector:

\langle b\left|\alpha\right\rangle = \sum\limits_a c_a \langle b\left|a\right\rangle = \sum\limits_a c_a \delta_{ab} = c_b
⟨b∣α⟩=∑aca⟨b∣a⟩=∑acaδab=cb\langle b\left|\alpha\right\rangle = \sum\limits_a c_a \langle b\left|a\right\rangle = \sum\limits_a c_a \delta_{ab} = c_b

Completeness

\left|\alpha\right\rangle = \sum\limits_ac_a\left|a\right\rangle
∣α⟩=∑aca∣a⟩\left|\alpha\right\rangle = \sum\limits_ac_a\left|a\right\rangle

Since 

c_a = \langle a\left|\alpha\right\rangle
ca=⟨a∣α⟩c_a = \langle a\left|\alpha\right\rangle

We see that the middle operator is basically a unity operator:

and

\left|\alpha\right\rangle = \sum\limits_a\left|a\right\rangle \langle a \left|\alpha\right\rangle
∣α⟩=∑a∣a⟩⟨a∣α⟩\left|\alpha\right\rangle = \sum\limits_a\left|a\right\rangle \langle a \left|\alpha\right\rangle
\left|\alpha\right\rangle = (\sum\limits_a\left|a\right\rangle \langle a \left|) \cdot|\alpha\right\rangle
∣α⟩=(∑a∣a⟩⟨a∣)⋅∣α⟩\left|\alpha\right\rangle = (\sum\limits_a\left|a\right\rangle \langle a \left|) \cdot|\alpha\right\rangle
\sum\limits_a\left|a\right\rangle \langle a | = 1
∑a∣a⟩⟨a∣=1\sum\limits_a\left|a\right\rangle \langle a | = 1

Projections and Probabilities

\langle \alpha \left|\alpha\right\rangle = \langle \alpha \left| (\sum_a|a\rangle \langle a|) |\alpha\right\rangle = \sum_a | \langle a|\alpha\rangle |^2
⟨α∣α⟩=⟨α∣(∑a∣a⟩⟨a∣)∣α⟩=∑a∣⟨a∣α⟩∣2\langle \alpha \left|\alpha\right\rangle = \langle \alpha \left| (\sum_a|a\rangle \langle a|) |\alpha\right\rangle = \sum_a | \langle a|\alpha\rangle |^2
\sum\limits_a |c_a|^2 = 1
∑a∣ca∣2=1\sum\limits_a |c_a|^2 = 1

Define:

\Lambda_a = |a\rangle \langle a|
Λa=∣a⟩⟨a∣\Lambda_a = |a\rangle \langle a|

Then:

\sum\limits_a \Lambda_a = 1
∑aΛa=1\sum\limits_a \Lambda_a = 1

Quantum Measurement

Dirac: "A measurement causes a state to jump into an eigenstate of the dynamical variable being measured"

\left|\alpha\right\rangle = \sum_a c_a \left|a\right\rangle \rightarrow \left|a\right\rangle
∣α⟩=∑aca∣a⟩→∣a⟩\left|\alpha\right\rangle = \sum_a c_a \left|a\right\rangle \rightarrow \left|a\right\rangle
\left|a\right\rangle \rightarrow \left|a\right\rangle
∣a⟩→∣a⟩\left|a\right\rangle \rightarrow \left|a\right\rangle

Probability for a: 

\sum\limits_a |c_a|^2
∑a∣ca∣2\sum\limits_a |c_a|^2

Selective Measurements

\left|a\right\rangle
∣a⟩\left|a\right\rangle

Only allow a single eigenstate through

\Lambda_a |\beta\rangle = |a\rangle
Λa∣β⟩=∣a⟩\Lambda_a |\beta\rangle = |a\rangle
|\beta\rangle
∣β⟩|\beta\rangle

Incompatible Observables

\left|b\right\rangle
∣b⟩\left|b\right\rangle
\left|a\right\rangle
∣a⟩\left|a\right\rangle
\left|c\right\rangle
∣c⟩\left|c\right\rangle

Groups of operators that you can't measure at the same time are called incompatible.

p(a \rightarrow c) = |\langle c | b \rangle |^2 |\langle b | a \rangle|^2
p(a→c)=∣⟨c∣b⟩∣2∣⟨b∣a⟩∣2p(a \rightarrow c) = |\langle c | b \rangle |^2 |\langle b | a \rangle|^2

Sum Over Intermediate States

\left|b\right\rangle
∣b⟩\left|b\right\rangle
\left|a\right\rangle
∣a⟩\left|a\right\rangle
\left|c\right\rangle
∣c⟩\left|c\right\rangle
p(a \rightarrow \forall b \rightarrow c) = \sum_b |\langle c | b \rangle |^2 |\langle b | a \rangle|^2
p(a→∀b→c)=∑b∣⟨c∣b⟩∣2∣⟨b∣a⟩∣2p(a \rightarrow \forall b \rightarrow c) = \sum_b |\langle c | b \rangle |^2 |\langle b | a \rangle|^2

A Final Twist

\left|a\right\rangle
∣a⟩\left|a\right\rangle
\left|c\right\rangle
∣c⟩\left|c\right\rangle
p(a \rightarrow \rightarrow \rightarrow c) = |\langle c | a \rangle |^2 = |\sum_b \langle c | b \rangle \langle b | a \rangle|^2
p(a→→→c)=∣⟨c∣a⟩∣2=∣∑b⟨c∣b⟩⟨b∣a⟩∣2p(a \rightarrow \rightarrow \rightarrow c) = |\langle c | a \rangle |^2 = |\sum_b \langle c | b \rangle \langle b | a \rangle|^2

Somehow don't observe the "b" filter (it's damaged, someone removed it, etc.)

|\sum_b \langle c | b \rangle \langle b | a \rangle|^2 \ne \sum_b |\langle c | b \rangle |^2 |\langle b | a \rangle|^2!
∣∑b⟨c∣b⟩⟨b∣a⟩∣2≠∑b∣⟨c∣b⟩∣2∣⟨b∣a⟩∣2!|\sum_b \langle c | b \rangle \langle b | a \rangle|^2 \ne \sum_b |\langle c | b \rangle |^2 |\langle b | a \rangle|^2!
\left|a\right\rangle = \sum\limits_b|b\rangle \langle b\left|a\right\rangle
∣a⟩=∑b∣b⟩⟨b∣a⟩\left|a\right\rangle = \sum\limits_b|b\rangle \langle b\left|a\right\rangle

Each b value measured.

b values unmeasured, left in superposition.

The Mystery of Quantum Mechanics

The result coming from the "c" filter depends on whether or not measurements on the "b" filter were carried out.

This is a truly quantum phenomenon, and is in many ways the essence of the field.

The two expression become equal if the A measurement and the B measurements are compatible, or if the B and C measurements are compatible.

Further Reading

I pretty much copied the first half of the first chapter of J. J. Sakurai's Modern Quantum Mechanics.

 

What we didn't cover:

Time evolution of quantum states (Hamiltonians, Schroedinger equation)

Incredible historical development of QM from about 1900s to 1920s

Pretty much everything.

Quantum Mechanics Is Weird

By chrislambda

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Quantum Mechanics Is Weird

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