Phy5646

Second Quantization

Consider now a wavefunction pertaining to a many-particle system, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi (\eta_1, \eta_2, \eta_3, ... , \eta_N) \!} , which is considered to be a field variable. For the many-particle system, this field variable must also quantized by a process known as second quantization.

In order to perform this quantization of the field variable, we must construct special raising and lowering operators, associated with the individual energy levels of the system, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}_j ^{\dagger} \!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}_j \!} , which add and subtract particles from the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{th} \!} energy level, respectively. In the presence of spin, an additional subscript is added to separate the creation and annihilation operators for each case of spin, so that each operator only acts on particles with the same spin attributed to said operator. In the simple, although rather non-physical, case of spinless particles, this extra factor can be ignored for simplicity in examining how the operators work on the quantized field:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}_j ^{\dagger} |0\rangle = |1\rangle \!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}_j |1\rangle = |0\rangle \!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}_j |0\rangle = 0 \!}

For the case of fermions, an additional constraint on the operators is placed due to the exclusion principle: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a} ^{\dagger} |1\rangle = 0 \!}

Given the two classes of particles, fermions and bosons, two sets of relations result to relate the creation and annihilation operators.

For the case of bosons, the operators obey a commutator relationship of the form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\hat{a}_i, \hat{a}_j ^{\dagger}] = \delta_{ij}; [\hat{a}_i, \hat{a}_j] = [\hat{a}_i ^{\dagger}, \hat{a}_j ^{\dagger}] = 0 \!}

The state of the system, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |n_0, n_1, .., n_N\rangle \!} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n_j \!} refers to the number of particles in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j^{th} \!} state, is therefore of the form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |n_0, n_1, .., n_N\rangle = {\frac{(\hat{a}_0 ^{\dagger})^{n_o}}{\sqrt{n_0 !}}}{\frac{(\hat{a}_1 ^{\dagger})^{n_1}}{\sqrt{n_1 !}}}...{\frac{(\hat{a}_N ^{\dagger})^{n_N}}{\sqrt{n_N !}}} |0\rangle \!}

Fermions, however, obey anti-commutator relationships, of the following form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ \hat{a}_i, \hat{a}_j ^{\dagger} \} = \delta_{ij}; \{ \hat{a}_i, \hat{a}_j \} = \{ \hat{a}_i ^{\dagger}, \hat{a}_j ^{\dagger} \} = 0 \!}

For this type of system, the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |n_0, n_1, .., n_N\rangle \!} can be written as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |n_0, n_1, .., n_N\rangle = (\hat{a}_0 ^{\dagger})^{n_0}(\hat{a}_1 ^{\dagger})^{n_1}...(\hat{a}_N ^{\dagger})^{n_N} |0\rangle \!}

Furthermore, for both classes of particles, we can create an operator that, upon acting on the total state of the system, returns the number of particles in a given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n^{th} \!} state (for fermions this will obviously be 0 or 1). This operator is of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}_n ^{\dagger} \hat{a}_n \!} . Therefore, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a}_j ^{\dagger} \hat{a}_j |n_0, n_1, .., n_N\rangle = n_j |n_0, n_1, .., n_N\rangle \!} . From this it is easy to obtain an operator, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{N} \!} that returns the total number of particles in the system:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{N} = \sum_{j=0}^{\infty} \hat{a}_n ^{\dagger} \hat{a}_n \!}

From this operator, the average number of particles, and therefore the average flux number, can be calculated by the following:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \Psi | \hat{N} | \Psi\rangle = \langle \hat{N} \rangle \!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \Psi | {\langle \hat{N} - {\langle \hat{N} \rangle} \rangle}^2 | \Psi\rangle = \langle \hat{N}^2 \rangle - {\langle \hat{N} \rangle}^2 \!}

To continue the analysis of these second quantization operators, lets consider the projection of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{a} ^{\dagger} \!} on the state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0\rangle \!} , that is to say, define a function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_j(\mathbf r) \!} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle \mathbf{r} | \hat{a}_j ^{\dagger} |0\rangle = \phi_j(\mathbf r) \!}

Rearranging this yields an expression for the position state of a particular particle, weighted over the different energy levels by the function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_j(\mathbf r) \!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\mathbf{r}\rangle = \sum_{j=0}^{\infty} \phi_j^*(\mathbf r)\hat{a}_j ^{\dagger} |0\rangle \!}

This new field operator acting on the ground state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |0\rangle \!} is called the field creation operator:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi ^{\dagger}(\mathbf r) \equiv \sum_{i=0}^{\infty} \phi_i^*(\mathbf r) \hat{a}_i ^{\dagger} \!}

Therefore, the position state of a system containing Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n\!} particles can be expressed as the following:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\mathbf{r}_1, \mathbf{r}_2, .., \mathbf{r}_n\rangle = {\frac{1}{\sqrt{n!}}}\Psi ^{\dagger}(\mathbf{r}_n)...\Psi ^{\dagger}(\mathbf{r}_2)\Psi ^{\dagger}(\mathbf{r}_1)|0\rangle\!}