Lab Methods - Graph

In laboratory experiments, graph are employed to determine how one measured variable depends on another. These graphs generally fall into three categories: linear, semilog (logarithmic versus linear), and log-log. Which type of graph listed in the third column below would NOT be the best for plotting data to test the relationship given in the first and second columns?

		Relation		Variables Plotted		Type of graph
A.		dN/dt ∝ e−2t		Activity vs time for a radioactive isotope		Semilog
B.		eVs = hf  − W		Stopping potential vs frequency for the photoelectric effect		Linear
C.		s ∝ t2		Distance vs time for an object undergoing constant acceleration		Log-log
D.		Vout/Vin ∝ 1/ω		Gain vs frequency for a low-pass filter		Linear
E.		P ∝ T4		Power radiated vs temperature for blackbody radiation		Log-log

(GR9677 #27)

Solution:

(A) dN/dt ∝ e^−2t
log [dN/dt] ∝ log e^−2t
log [dN/dt] ∝ −2t
Activity vs time graph has a semilog plot.

(B) V_s ∝ f
Stopping potential vs frequency graph has a linear plot.

(C) s ∝ t²
log s ∝ 2 log t
Distance vs time graph has a log-log plot.

(D) V_out/V_in ∝ 1/ω
V_out/V_in ∝ ω⁻¹
log [V_out/V_in] ∝ − log ω
Gain vs frequency graph has a log-log plot, NOT linear.

(E) P ∝ T⁴
log P ∝ 4 log T
Power vs temperature graph has a log-log plot.

Answer: D

Notes:

A semilogarithmic graph has one axis in logarithmic scale and the other in linear scale.

A log-log graphs has both axes in logarithmic scale.

Quantum Mechanics - Wave Function

If a freely moving electron is localized in space to within ∆x₀ of x₀, its wave function can be described by a wave packet

where f(k) is peaked around a central value k₀. Which of the following is most nearly the width of the peak in k?

A. 

B.

C.

D.

E.

(GR9277 #27)

Solution:

In quantum mechanics, the momentum p = ħk and position x wave functions are Fourier transform pairs and the relation between p and x representations forms the Heisenberg uncertainty relation:

∆x∆k ≥ 1 → ∆k ≥ 1/∆x

Or, since k and x are fourier variables, their localization would vary inversely.

Answer: B

Nuclear & Particle Physics - Spin

The hypothesis that an electron possesses spin is qualitatively significant for the explanation of all the following topics EXCEPT the

A. Structure of the periodic table
B. Specific heat of metals
C. Anomalous Zeeman effect
D. Deflection of moving electron by a uniform magnetic field
E. Fine structure of atomic spectra

(GR8677 #27)

Solution:

(A) TRUE
Spin → Pauli exclusion principle → electron configuration → Structure of the periodic table

(B) TRUE
Specific heat for Fermions (½-integer spin) is different from Bosons (integer spin).

(C) TRUE
Zeeman Effect: the splitting of spectral lines when an external magnetic field is applied.
"Normal" Zeeman effect → This type of splitting is observed for spin 0 states since the spin does not contribute to the angular momentum.
"Anomalous" Zeeman effect → When electron spin is included, there is a greater variety of splitting patterns.

(D) FALSE
Deflection of moving electron by a uniform magnetic field does not depend on spin

(E) TRUE
Fine structure = the splitting of the spectral lines of atoms due to quantum mechanical (electron spin) and relativistic corrections.

Answer: D

Quantum Mechanics - Hermitian Operator

The eigenvalues of a Hermitian operator are always

A. Real
B. Imaginary
C. Degenerate
D. Linear
E. Positive

(GR0177 #27)

Solution:

The eigenvalues are always real and hence observable.
Proof:


Hermitian operator: .
So,   real

Answer: A