Course: PHYS 5426 — Quantum Statistical Physics Assignment #1 Instructor: Gennady Y. Chitov Date Assigned: January 15, 2014 Due Date: January 29, 2014 Problem 1. Prove [a; f(a†)] = @f(a†) @a† (1) [a†; f(a)] = −@f(a) @a (2) for arbitrary function f of operator which admits a series expansion. The Bose creation/ annihilation operators satisfy the standard commutation relations [a; a†] ≡ aa† − a†a = 1 (3) Hint: From Eqs.(1,2) one can figure out the corresponding commutation relations for the powers of creation/annihilation operators and then prove them by the method of mathematical induction. Note that for an arbitrary operator Aˆ: @A^n @A^ = nAˆn−1. Problem 2. In the presence of a constant external force acting on a one-dimensional oscillating particle its Hamiltonian becomes that of the so-called displaced oscillator, and the Schr¨odinger equation ˆH (q) = E (q) of the problem (cf. lecture notes) can be written in terms of dimensionless variables as ( − 1 2 d2 d2 + 1 2 2 − √ 2  ) () = ” () ; (4) where q = √ ~ m! and E = ~!”. a). Write the Schr¨odinger equation (4) in terms of the creation/annihilation operators of the harmonic oscillator ( = 0)  = √1 2 (a + a†) (5) d d = √1 2 (a − a†) (6) 1 Via a linear transformation to the new creation/annihilation operators ˜a†; ˜a preserving the bosonic commutation relations for ˜a†; ˜a map the problem (4) of the displaced oscillator onto that of a simple harmonic oscillator with new operators (˜a†; ˜a). b). Find the spectrum (eigenvalues) ” (E) of the displaced oscillator. c). Write the normalized eigenstates |n⟩ of the displaced Hamiltonian (4) via a† and the vacuum state |Θ◦⟩ of the new operators, i.e. ˜a|Θ◦⟩ = 0 (7) d). As follows from the completeness of the oscillator’s eigenstates, the vacuum state of the displaced oscillator |Θ◦⟩ can be related to the simple oscillator’s vacuum |0⟩ (i.e., a|0⟩ = 0) as |Θ◦⟩ = Ω(a†)|0⟩ (8) Find (up to a normalization factor) the operator function Ω(a†) relating two vacua. Hint: in working out Eqs.(7,8), employ Eqs.(1,2). Problem 3. Prove from the standard commutation relations ([ai; a † j ]∓ = ij , etc) that ⟨0|aiaja † ka † l |0⟩ = jkil ± ikjl (9) the sign depending on the statistics. Also calculate the vacuum expectation value ⟨0|ahaiaja † ka † l a† m |0⟩. Problem 4. In the formalism of second quantization the two-particle interaction term of the Hamiltonian for spinless fermions is given by ˆ V = 1 2 ∫ ∫ dxdy ˆ †(x) ˆ †(y)V(x; y) ˆ (y) ˆ (x) (10) For the short-ranged interaction V(x; y) = V(|x−y|) ≡ V(r) = e2 exp(−r)=r find ˆ V in the momentum representation. The field operators and the creation/annihilation operators in the momentum representation are related in the usual way, i.e., ˆ †(x) = ∫ dp (2)3 a†(p)e−ipx (11) Note that the limit  → 0 recovers the Coulomb (long-ranged) interaction V(r) = e2=r. What is the Fourier transform V(q) of the Coulomb interaction? 2 Problem 5. The matrix elements of a two-particle interaction from the previous problem can be written as ⟨k3k4|V|k1k2⟩ = (2)3(k1 + k2 − k3 − k4)V(q) (12) where q ≡ k3−k1 is the momentum transfer. Show that the diagonal part of the interaction operator ˆ V found on the previous problem in the k-representation, arises from momentum transfers q = 0 and q = k2−k1. Write down the two interaction terms and identify them as direct (q = 0) and exchange (q = k2 − k1) interactions. Draw the corresponding Feynman diagrams. Problem 6. Find the first correction to the temperature dependence of the chemical potential  of the degenerate ideal electron gas, assuming constant particle concentration ⟨N⟩=V . Express the result in terms of T and the zero-temperature chemical potential ◦. For the calculations the following formula (we set kB = 1) can be used: I ≡ ∫ ∞ 0 f(“)d” e(“−)=T + 1 = ∫  0 f(“)d” + 2 6 T2f′() + O(T4) (13) 3

Course: PHYS 5426 — Quantum Statistical Physics Assignment #1 Instructor: Gennady Y. Chitov Date Assigned: January 15, 2014 Due Date: January 29, 2014 Problem 1. Prove [a; f(a†)] = @f(a†) @a† (1) [a†; f(a)] = −@f(a) @a (2) for arbitrary function f of operator which admits a series expansion. The Bose creation/ annihilation operators satisfy the standard commutation relations [a; a†] ≡ aa† − a†a = 1 (3) Hint: From Eqs.(1,2) one can figure out the corresponding commutation relations for the powers of creation/annihilation operators and then prove them by the method of mathematical induction. Note that for an arbitrary operator Aˆ: @A^n @A^ = nAˆn−1. Problem 2. In the presence of a constant external force acting on a one-dimensional oscillating particle its Hamiltonian becomes that of the so-called displaced oscillator, and the Schr¨odinger equation ˆH (q) = E (q) of the problem (cf. lecture notes) can be written in terms of dimensionless variables as ( − 1 2 d2 d2 + 1 2 2 − √ 2  ) () = ” () ; (4) where q = √ ~ m! and E = ~!”. a). Write the Schr¨odinger equation (4) in terms of the creation/annihilation operators of the harmonic oscillator ( = 0)  = √1 2 (a + a†) (5) d d = √1 2 (a − a†) (6) 1 Via a linear transformation to the new creation/annihilation operators ˜a†; ˜a preserving the bosonic commutation relations for ˜a†; ˜a map the problem (4) of the displaced oscillator onto that of a simple harmonic oscillator with new operators (˜a†; ˜a). b). Find the spectrum (eigenvalues) ” (E) of the displaced oscillator. c). Write the normalized eigenstates |n⟩ of the displaced Hamiltonian (4) via a† and the vacuum state |Θ◦⟩ of the new operators, i.e. ˜a|Θ◦⟩ = 0 (7) d). As follows from the completeness of the oscillator’s eigenstates, the vacuum state of the displaced oscillator |Θ◦⟩ can be related to the simple oscillator’s vacuum |0⟩ (i.e., a|0⟩ = 0) as |Θ◦⟩ = Ω(a†)|0⟩ (8) Find (up to a normalization factor) the operator function Ω(a†) relating two vacua. Hint: in working out Eqs.(7,8), employ Eqs.(1,2). Problem 3. Prove from the standard commutation relations ([ai; a † j ]∓ = ij , etc) that ⟨0|aiaja † ka † l |0⟩ = jkil ± ikjl (9) the sign depending on the statistics. Also calculate the vacuum expectation value ⟨0|ahaiaja † ka † l a† m |0⟩. Problem 4. In the formalism of second quantization the two-particle interaction term of the Hamiltonian for spinless fermions is given by ˆ V = 1 2 ∫ ∫ dxdy ˆ †(x) ˆ †(y)V(x; y) ˆ (y) ˆ (x) (10) For the short-ranged interaction V(x; y) = V(|x−y|) ≡ V(r) = e2 exp(−r)=r find ˆ V in the momentum representation. The field operators and the creation/annihilation operators in the momentum representation are related in the usual way, i.e., ˆ †(x) = ∫ dp (2)3 a†(p)e−ipx (11) Note that the limit  → 0 recovers the Coulomb (long-ranged) interaction V(r) = e2=r. What is the Fourier transform V(q) of the Coulomb interaction? 2 Problem 5. The matrix elements of a two-particle interaction from the previous problem can be written as ⟨k3k4|V|k1k2⟩ = (2)3(k1 + k2 − k3 − k4)V(q) (12) where q ≡ k3−k1 is the momentum transfer. Show that the diagonal part of the interaction operator ˆ V found on the previous problem in the k-representation, arises from momentum transfers q = 0 and q = k2−k1. Write down the two interaction terms and identify them as direct (q = 0) and exchange (q = k2 − k1) interactions. Draw the corresponding Feynman diagrams. Problem 6. Find the first correction to the temperature dependence of the chemical potential  of the degenerate ideal electron gas, assuming constant particle concentration ⟨N⟩=V . Express the result in terms of T and the zero-temperature chemical potential ◦. For the calculations the following formula (we set kB = 1) can be used: I ≡ ∫ ∞ 0 f(“)d” e(“−)=T + 1 = ∫  0 f(“)d” + 2 6 T2f′() + O(T4) (13) 3

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Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 Assignment 4 – Noise and Correlation 1. If a signal is measured as 2.5 V and the noise is 28 mV (28 × 10−3 V), what is the SNR in dB? 2. A single sinusoidal signal is found with some noise. If the RMS value of the noise is 0.5 V and the SNR is 10 dB, what is the RMS amplitude of the sinusoid? 3. The file signal_noise.mat contains a variable x that consists of a 1.0-V peak sinusoidal signal buried in noise. What is the SNR for this signal and noise? Assume that the noise RMS is much greater than the signal RMS. Note: “signal_noise.mat” and other files used in these assignments can be downloaded from the content area of Brightspace, within the “Data Files for Exercises” folder. These files can be opened in Matlab by copying into the active folder and double-clicking on the file or using the Matlab load command using the format: load(‘signal_noise.mat’). To discover the variables within the files use the Matlab who command. 4. An 8-bit ADC converter that has an input range of ±5 V is used to convert a signal that ranges between ±2 V. What is the SNR of the input if the input noise equals the quantization noise of the converter? Hint: Refer to Equation below to find the quantization noise: 5. The file filter1.mat contains the spectrum of a fourth-order lowpass filter as variable x in dB. The file also contains the corresponding frequencies of x in variable freq. Plot the spectrum of this filter both as dB versus log frequency and as linear amplitude versus linear frequency. The frequency axis should range between 10 and 400 Hz in both plots. Hint: Use Equation below to convert: Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 6. Generate one cycle of the square wave similar to the one shown below in a 500-point MATLAB array. Determine the RMS value of this waveform. [Hint: When you take the square of the data array, be sure to use a period before the up arrow so that MATLAB does the squaring point-by-point (i.e., x.^2).]. 7. A resistor produces 10 μV noise (i.e., 10 × 10−6 V noise) when the room temperature is 310 K and the bandwidth is 1 kHz (i.e., 1000 Hz). What current noise would be produced by this resistor? 8. A 3-ma current flows through both a diode (i.e., a semiconductor) and a 20,000-Ω (i.e., 20-kΩ) resistor. What is the net current noise, in? Assume a bandwidth of 1 kHz (i.e., 1 × 103 Hz). Which of the two components is responsible for producing the most noise? 9. Determine if the two signals, x and y, in file correl1.mat are correlated by checking the angle between them. 10. Modify the approach used in Practice Problem 3 to find the angle between short signals: Do not attempt to plot these vectors as it would require a 6-dimensional plot!

Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 Assignment 4 – Noise and Correlation 1. If a signal is measured as 2.5 V and the noise is 28 mV (28 × 10−3 V), what is the SNR in dB? 2. A single sinusoidal signal is found with some noise. If the RMS value of the noise is 0.5 V and the SNR is 10 dB, what is the RMS amplitude of the sinusoid? 3. The file signal_noise.mat contains a variable x that consists of a 1.0-V peak sinusoidal signal buried in noise. What is the SNR for this signal and noise? Assume that the noise RMS is much greater than the signal RMS. Note: “signal_noise.mat” and other files used in these assignments can be downloaded from the content area of Brightspace, within the “Data Files for Exercises” folder. These files can be opened in Matlab by copying into the active folder and double-clicking on the file or using the Matlab load command using the format: load(‘signal_noise.mat’). To discover the variables within the files use the Matlab who command. 4. An 8-bit ADC converter that has an input range of ±5 V is used to convert a signal that ranges between ±2 V. What is the SNR of the input if the input noise equals the quantization noise of the converter? Hint: Refer to Equation below to find the quantization noise: 5. The file filter1.mat contains the spectrum of a fourth-order lowpass filter as variable x in dB. The file also contains the corresponding frequencies of x in variable freq. Plot the spectrum of this filter both as dB versus log frequency and as linear amplitude versus linear frequency. The frequency axis should range between 10 and 400 Hz in both plots. Hint: Use Equation below to convert: Biomedical Signal and Image Processing (4800_420_001) Assigned on September 12th, 2017 6. Generate one cycle of the square wave similar to the one shown below in a 500-point MATLAB array. Determine the RMS value of this waveform. [Hint: When you take the square of the data array, be sure to use a period before the up arrow so that MATLAB does the squaring point-by-point (i.e., x.^2).]. 7. A resistor produces 10 μV noise (i.e., 10 × 10−6 V noise) when the room temperature is 310 K and the bandwidth is 1 kHz (i.e., 1000 Hz). What current noise would be produced by this resistor? 8. A 3-ma current flows through both a diode (i.e., a semiconductor) and a 20,000-Ω (i.e., 20-kΩ) resistor. What is the net current noise, in? Assume a bandwidth of 1 kHz (i.e., 1 × 103 Hz). Which of the two components is responsible for producing the most noise? 9. Determine if the two signals, x and y, in file correl1.mat are correlated by checking the angle between them. 10. Modify the approach used in Practice Problem 3 to find the angle between short signals: Do not attempt to plot these vectors as it would require a 6-dimensional plot!

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