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Quantum mechanics describes the microscopic properties of nature in a regime where classical mechanics no longer applies. It explains phenomena such as the wave-particle duality, quantization of energy, and the uncertainty principle and is generally used in single-body systems. Use the quantum-field-theory tag for the theory of many-body quantum-mechanical systems.

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Finite Potential Barrier Where $E=V_0$

If we have a free particle of energy E incident on a potential $V$ $$V(x) = \begin{cases}0 & x \leq 0 \\ V_0 & 0 < x < L \\ 0 & x \geq L\end{cases}$$ We find that the wave function $\phi$ $$\phi(x) = …
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1 answer
30 views

How is the measurement of the $Z_0Z_2$ error syndrome realised by these two circuits?

I am told that the following quantum circuit diagram implements the $Z_0Z_2$ error syndrome, but I don't follow. I understand error syndromes when expressed in matrix form and how performing $Z_0Z_2| …
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0 votes
1 answer
130 views

Why must normalisable eigenfunctions have $E > V_{min}$? [duplicate]

I have read normalisable eigenfunctions of the Hamiltonian operator. $$\hat{H}\phi = E\phi$$ If $\phi$ is to be normalisable we must have $E > V_{min}$ Why is this?
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1 answer
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Derivation of QFT product formula

If the quantum fourier transform is defined as follows: $$ U_{FT} |x\rangle = \frac{1}{2^{n/2}} \sum^{2^n-1}_{y=0} e^{2\pi i x y / 2^n} | y \rangle. $$ We can rewrite the exponential term as: $$ e^{2\ …
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3 votes
1 answer
707 views

Understanding the probability of measurement w.r.t. density matrix

I am told that the probability of measuring $\lambda$ is $$p_\lambda = Tr(\hat{P}_\lambda\hat{\rho}) = Tr(\hat{P}_\lambda\hat{\rho}\hat{P}_\lambda)$$ where $\hat{P}_\lambda = \sum_{n:\lambda_n = \lamb …
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11 votes
2 answers
2k views

Why is the trace of the outer product of two states equal to the inner product of the two st...

Why is it that, given two quantum states $|\psi_1\rangle$, $|\psi_2\rangle$, $$\mathrm{Tr}(|\psi_1\rangle\langle\psi_2|) = \langle\psi_2|\psi_1\rangle \quad $$ I went through the equation with the qub …
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